# Unfair Coin Probability

p is the probability of. A probability of one means that the event is certain. are within the probability for a fair coin. Suppose I have an unfair coin and I want to turn it into a fair coin using the following way, Probability of generating head is equal for unfair coin; Flip unfair coin and only accept head; When a head is appearing, treat it as 1 (head for virtual fair coin), when another head is appearing, treat it as 0 (tail for virtual fair. If the coin is flipped 14 times, what is the probability that;a) it comes up tails exactly 6 times?;b) it comes up heads more than 11 times?. Coin A has a 90% chance of heads, coin B has a 5% chance heads. Simulating fair coins with unfair coins (and vice versa) → Flipping HHT before HTT? Not what you think. khanacademy. But again, it was possible that the coin had been fair. In This Part: Fair and Unfair Allocations The average for a set of data corresponds to the equal-shares allocation or fair allocation of the data. What is the probability that B got more heads than A? A fair coin is tossed until a head comes up for first time. In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. The theory of probability has always been associated with gambling and many most accessible examples still come from that activity. , n Bernoulli trials). If the coin is tossed 3 times, what is the probability that at least one of the tosses will turn up tails?. Students often respond initially that the game is unfair because the player who wins when the coins match has two chances to win while the other player only has one chance. Now that in and of itself is not proof, so. And the probability that no one wins: Pr(no one wins) = 0. Case 2: One head. WEAR Abstract. If we see a coin tossed twice and we see 2 heads, we'd like to know if the coin is fair, or at least to be able to determine the probability that the coin is fair. In the preface, Feller wrote about his treatment of uctuation in coin tossing: \The results are so amazing and so at variance with common intuition that even sophisticated colleagues doubted that coins actually misbehave as theory predicts. If you know how to manage time then you will surely do great in your. In this paper, we explore some properties of the cumulative probability distribution of this random variable. Read about me, or email me. coin toss probability calculator,monte carlo coin toss trials. An example of such a vector for three possible outcomes is c(0. The standard (maybe overused) example is flipping a fair coin. When I flip the coin and get tails, I lose a dollar. 24 to estimate a probability of 0. After all, the probability that you will roll HHT right off the bat is , and the same for HTT and HHH, so there should be symmetry. Probability (unfair coin) Hello everyone, I need a little help with this problem I have. The remaining coins have heads on both sides. Two people take turns to ip a coin, Your opponent starts rst, you second. A common topic in introductory probability is solving problems involving coin flips. Not convinced? Suppose I give you a dollar for every head, and you give me a dollar for every tail, and we ip the coin 10 times (my coin!) and we get 0 heads and 10 tails. 9%: an unfair coin lands heads 65% of the time. Determining Dependent and Independent Events (HSS-CP. Read more about setting a seed below. The appeal of the coin toss that it is a simple, seemingly unbiased, method of deciding between 2 options. (b) What is the chance that the coin is flipped exactly $$i$$ times? (c) What is the chance that the coin is flipped more than twice? (d) Repeat the previous three questions for a unfair coin which has probability $$p$$ of getting Tails. In the case of coins, heads and tails each have the same probability of 1/2. Inequalities with Absolute Values. If you toss this coin twice, what is the pro… Get the answers you need, now!. In this case A is flipping 10 heads in a row and B is picking the two-headed coin. The remaining coins have heads on both sides. a) How to make an event with 50% probability? b) Expected number of flips until a realization occurs? c) Can you create a strategy to reduce the number of flips necessary? d) Can you create a strategy to reduce the number of flips necessary for an unfair coin with any bias?. With H0 = "coin is unfair", H1= "coin is fair", and S = "test succesful with p-value < 0. ) the probability that a coin flip will result in heads (set to a default of 0. Students often respond initially that the game is unfair because the player who wins when the coins match has two chances to win while the other player only has one chance. Otherwise, a student from a different class containing 12 boys and 9 girls is selected. Coin A has a 90% chance of coming up heads, coin B has a 5% chance of coming up heads. This is a common misconception that is best addressed through data collection and analyzing that data rather than through telling. What is the probability of getting 4 tails in 4 tosses of an unfair coin where probability of tails is 7?. But, could not arrive at a solution. 7 probability of landing 'heads'. Active 2 years, 7 months ago. Consider an unfair coin. 6 and tails with probability 0. But diving straight into Huszár (2017) or Chen et al (2017) can be a challenge, especially if you're not familiar with the basic concepts and underlying math. PROBABILITY SPACE A probability space has three components: (1) A sample space Ω that is the set of all possible outcomes of the random process modeled; (2) A family of sets F representing the allowable events , where each set in F is a subset of the sample space Ω; (3) A probability function. e a coin with equal probability of landing heads or tails) but would like to construct an outcome of biased probability , how would you do it? I remember this question. If you toss the coin 40 times, how many heads do you expect to see? A. An unfair coin is tossed; if a head appears uppermost, then a marble is selected from bag (X); otherwise, a marble is selected from bag (Y). An unfair coin has probability 0. Question 2 An insurance company writes policies for a large number of newly-licensed drivers each year. Find the probability of exactly one Heads. 4 y Expected Value Some Good Advice Pay careful attention to what notation tells you to do in performing a calculation. Build and represent graphically the probability distribution and the cumulative distribution of the function with random variable "X = number of time result is head". Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. We express probability as a number between 0 and 1. 4 is tossed three times. Increasing the repetitions, you can compare the paths taken in repea. What is the probability categorized under Math and Probability. A random variableXis: PSYCH 2000-Consider and unfair coin. How do these numerical results compare to actual outcomes? Under the current overtime rules, there have been 51 overtime games. Mathematically, we find probability by comparing the number of favorable outcomes to the number of possible outcomes. , 𝑝 ℎ𝑒𝑎𝑑𝑠 = 0. The formula of probability of an event G: P (G) = Discussion: 1) If the coin is biased or unfair, what happens to the probability of getting “Heads”?. With dice rolling, your sample space is going to be every possible dice roll. Design a game between Alice and Bob so that Alice's winning probability is exactly α. Each biased coin has a probability of a head 4/5. Let a and b be the results of 2 tosses of the unfair coin. Instead, you only find a biased coin with probability 1/3 of getting heads. Competitive acquisition plan for low probability of detection data link networks. Make a fair coin from a biased coin You are given a function foo() that represents a biased coin. The probability is 0. In the above experiment, we used a fair coin. Can someone explain to me how can you get a fair (equal probability) outcome using only an unfair coin (where unfair means that it will land head with probability p and tails 1-p where p !=. The game of course has to end in a ﬁnite number of tosses with probability 1. Lec -7 Frequency Probability and Unfair Coins. P(result of a coin toss is heads). At first glance this might seem like an overcomplicated way of solving this. In calcu-lating expected value, you are told to ﬁrst multiply the probability of each outcome by its. Here we will learn how to find the probability of tossing two coins. Hypothesis Testing 1 Hypothesis Testing Much of classical statistics is concerned with the idea of hypothesis testing. After you choose your first coin and flip it, you can base your decision of which coin to flip second on your results of the first flip. 7 is the probability of each choice we want, call it p. Luetkemeyer, Mr. If , discard observations, goto step 1. 1% probability that it will come up heads all ten times. The order does not matter as long as there are two head and two tails in the flip. This is a common misconception that is best addressed through data collection and analyzing that data rather than through telling. Example-Binomial #Suppose you have a biased coin that has a probability of 0. There is a die and a coin. For example, suppose we have three coins. Subscribe for more updates. Which answers, if any. PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS. Alice and Bob play a game as follows. The first two happen to Jackie with the same chance, but the third happens of the time, since the unfair coin is heads instead of tails. For example, suppose we have three coins. 2) - We look for all possible outcomes. Persi Diaconis has spent much of his life turning scams inside out. More generally, there are situations in which the coin is biased, so that heads and tails have different probabilities. , 𝑝 ℎ𝑒𝑎𝑑𝑠 = 0. If the coin is flipped 14 times, what is the probability that;a) it comes up tails exactly 6 times?;b) it comes up heads more than 11 times?. Output: Now, suppose we want to simulate 100 flips of an unfair two-sided coin. A random variableXis: PSYCH 2000-Consider and unfair coin. The Binomial Likelihood Function Note the similarity between the probability function and the likelihood function; the right hand sides are the same. Image Transcriptionclose. Probability Coin Flip Also, I've always had a problem with the oversimplified coin flip 50-50 thing. coin=randi([0:1], [100,1]) It should more or less give you 50 0's and 50 1's. Note that the. Dice Roll Probability for 6 Sided Dice: Sample Spaces. 9%: an unfair coin lands heads 65% of the time. If a fair coin (one with probability of heads equal to 1/2) is flipped a large number of times, the proportion of heads will tend to get closer to 1/2 as the number of tosses increases. Using Python 2. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. We're thinking about how the probability of an event can be dependent on another event occuring in this example problem. With dice rolling, your sample space is going to be every possible dice roll. 6 of landing heads. 25 " = 25% = 1/4 Probabilities are usually given as fractions. The coin is tossed four times. 7870 and the probability of getting three or more heads in a row or three or more tails in a row is 0. 2 is flipped. What about probabilities when we don't have equally likely events? Say, we have unfair coins? If you're seeing this message, it means we're having trouble loading external resources on our website. (Remember, to calculate probability when the question includes the word “and”, you multiply. The coin is flipped 50 times. What is the probability that you must flip the coin four or more times. Example The same unfair coin as in the previous example is ﬂipped three times. Note that the. Here you can assume that if a child is a girl, her name will be Lilia with probability $\alpha \ll 1$ independently from other children's names. Click to left of y-axis to for a new run, to right of y-axis to pause. This theorem will justify mathematically both our frequency concept of probability and the interpretation of expected value as the average value to be expected in a large number of experiments. (a) What is the probability that the flipped coin will come up heads? We'll assume there is an equal chance (1/3) of picking any of the three coins. Suppose I have an unfair coin, and the probability of flip a head (H) is p, probability of flip a tail (T) is (1-p). - 11296804. Drivers spend on it is unfair that something like that To speak to a pet-related charity In group quarters - homes for the book value (i Rosario marchese: sorry; we’re going the profit and costing for healthcare software architect jobs Mail wasn't being forwarded correctly Infographics, insurance rate - get discounts up to $15,000 per person. What is the sample space for a fair coin ip? For a sequence of three coin ips? For a sequence of ve coin ips in which at least four ips turn out to be heads? Suppose you are told now that the coin is unfair, with the probability of a toss resulting in heads being 0. Suppose an unfair coin (P(head)=. If you know how to manage time then you will surely do great in your. Let’s start with our original problem: using a fair coin to simulate a coin with P(H) = 1/3, or P(T) = 2/3. The probability of having the unfair coin is$\frac13$. The pencils come in packages of 6. I was a mathematician, and now work in finance (systematic trading). With H0 = "coin is unfair", H1= "coin is fair", and S = "test succesful with p-value < 0. Probability of compound events Learn how to calculate the probability of at least 2 simple events. For example, suppose that each of 9 people has several dollars and altogether they have$45. Coin Toss Probability Calculator is a free online tool that displays the probability of getting the head or a tail when the coin is tossed. Frequency Stability Example: Range and mid-range This original Khan Academy video was translated into isiZulu by Wazi Kunene. Would you modify your approach to the the way you test the fairness of coins?. As above we can use R to simulate an experiment of rolling a die a number of times and compare our results with the theoretical probability. The coin is flipped 50 times. Competitive exams are all about time. Consider an unfair coin. The multinomial law gives the probability of exactly ki occurrances of the i-th face in n tosses of a die. If the coin shows tails, we draw a marble from urn T with 4 red and 2 blue marbles. Although it is. A box contains 5 fair coins and 5 biased coins. 3 Discrete Distributions A discrete distribution assigns a probability to every atom in the sample space of a random variable. Coin flipping game: how to make a fair toss from an unfair coin. Repeat task 4, but for a biased die with a probability of 0. A coin is tossed and comes up tails ten times: is this just random chance, or is an unfair coin being used? Learn when to reject the null hypothesis: if the probability (P) is less than the chosen significance level, the null must be. A bag contains 5 coins. We can explore this problem with a simple function in python. Binomial Distribution based on an Unfair Coin. Can you design a game where you and your opponent have an equal chance of winning? Show Answer. Find the probability of getting 4 heads. A random variableXis: PSYCH 2000-Consider and unfair coin. Please try solving this problem before jumping on the solution Click to learn. A classic example of a probabilistic experiment is a fair coin toss, in which the two possible outcomes are heads or tails. What about probabilities when we don't have equally likely events? Say, we have unfair coins? If you're seeing this message, it means we're having trouble loading external resources on our website. The probability of having the unfair coin is $\frac13$. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. Question: Consider Two Coins, One Fair And One Unfair. Frequency Stability Example: Range and mid-range This original Khan Academy video was translated into isiZulu by Wazi Kunene. If the coin is tossed 3 times, what is the probability that at least 1 of the tosses will turn up tails? A. Students often respond initially that the game is unfair because the player who wins when the coins match has two chances to win while the other player only has one chance. If we have a biased coin (i. 12, 2012 Title 16 Commercial Practices Parts 0 to 999 Revised as of January 1, 2013 Containing a codification of documents of general applicability and future effect As of January 1, 2013. If we see a coin tossed twice and we see 2 heads, we'd like to know if the coin is fair, or at least to be able to determine the probability that the coin is fair. Determining Dependent and Independent Events (HSS-CP. ” “If that happens, then we’ll incorrectly decide that an unfair coin is really fair --called a ‘miss’. After all, the probability that you will roll HHT right off the bat is , and the same for HTT and HHH, so there should be symmetry. "Count line" can be moved by mouse. Well, that's the same as the probability that the first coin didn't come up heads when both coins are different, which is 1 - 1/2 = 1/2. Last time we talked about independence of a pair of outcomes, but we can easily go on and talk about independence of a longer sequence of outcomes. Pr[HHH] = Pr[TTT] = 1=2. ” “If that happens, then we’ll incorrectly decide that an unfair coin is really fair --called a ‘miss’. Download All Slides. What is the probability that a fair coin lands Heads 4 times out of 5 flips? Ans: C(5,4)/25 = 5/32. The coin is weighted so that the head {H} is 3 times more likely to occur than tails. When a coin is tossed, there lie two possible outcomes i. Consider the unfair coin with P(H) = 1/3 and P(T) = 2/3. So the probability of getting two heads is: 1 " in " 4 = 0. sim_fair_coin table (sim_fair_coin) Since there are only two elements in outcomes, the probability that we "flip" a coin and it lands heads is 0. The simplest case when you're learning to calculate dice probability is the chance of getting a specific number with one die. Taking many of the concepts he has covered in the last few videos, including probability, combinations, and conditional probability, Sal uses the example of fair and unfair coins in a bag to show various probability problems. Can you design a game where you and your opponent have an equal chance of winning? Show Answer. Design a game between Alice and Bob so that Alice's winning probability is exactly α. This is a formal framework that we can use to pose questions about a variety of topics in a consistent form that lets us apply statistical techniques to make statements about how results that we've gathered relate to questions that we're interested in. In fact, the probability for most other values virtually disappeared — including the probability of the coin being fair (Bias = 0. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. For this problem we can use the normal approximation to the binomial distribution. Create a function like flip_heads(), but for a fair die instead, and have it focus on the probability of rolling a 4. The order does not matter as long as there are two head and two tails in the flip. For example, an unfair coin could have p(X=h) = 0. In your simulation of flipping the unfair coin 100 times, how many flips came up heads? Include the code for sampling the unfair coin in your response. Probability : We have a weighted coin which shows a Head with probability p, (0. Let's write a function that takes in two arguments: 1. Quiz CHAPTER 16 NAME:_____ UNDERSTANDING PROBABILITY AND LONG-TERM EXPECTATIONS 1. These games work with random events, so they are a useful way to learn how to use probabilities to predict events. Then, the probability of heads is not 0. In fact Bayes theorem gives us a way to compute this value, but we need more information: priors. a) What is the probability that the coin comes up tails more than 25 times? b) What is the probability that the coin comes up tails more than 30 times?. Construct the probability distribution for x , and calculate its mean. For example, consider a fair coin. What is the probability that it lands heads at least once? Log On. HOWEVER, the question does noy say it is a fair coin!!!! The fact that it has landed 61 times on heads in 100 tosses could be because it is an "UNFAIR" coin and is weighted to favour heads compared to tails. ) please help!. Tossing an unfair coin multiple times. If you keep rolling a fair coin, what is the probability that you will get HHT before HTT (H = heads, T = tails)? How about HHH before HTT? HHT before TTT?. Suppose Tori has an unfair coin which lands on Tails with probability 0. 51), then we would expect that the results would yield 25. For an unfair or weighted coin, the two outcomes are not equally likely. WEAR Abstract. My initial idea is that we need to choose appropriate. 2 What is the. Since there are only two elements in outcomes, the probability that we "flip" a coin and it lands heads is 0. Probability: Tossing an Unfair Coin. Click here for how to write a probability. 1 Coin Tossing. 2 Prediction (EMG52) Games of chance (EMG53). In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. In some situations, such as in flipping an unfair coin, we cannot calculate the theoretical probability. You can load a die but you can't bias a coin Andrew Gelman∗ and Deborah Nolan† April 26, 2002 Abstract Dice can be loaded—that is, one can easily alter a die so that the probabilities of landing on the six sides are dramatically unequal. I'm a beginner with R and I am trying to design a coin flip simulation. Suppose you have an extremely unfair die: The probability of a 6 is 3/8, and the probability of each other number is 1/8. Knowing a little bit about the laws of probability, I quickly knew the fraction "2/6" for two dice and "3/6" for three dice was incorrect and spent a brief moment computing and then explaining the true percentages. Construct a probability model for this experiment. 9%: an unfair coin lands heads 65% of the time. If you toss a coin, it will come up a head or a tail. Please try solving this problem before jumping on the solution Click to learn. equals the probability that the expectation of a coin flip. B)If The Coin Is Flipped 5 Times, What Is The Probability Of Getting Exactly 2 Tails?. ) the number of games to be played, and 2. If he flips the coin three times, what is the probability that he flips more Heads than Tails? Express your answer as a common fraction. For a single toss of a balanced coin, let x = 1 for a head and x = 0 for a tail. So, after 500 flips most of the probability gets distributed around the value 0. Variational inference is all the rage these days, with new interesting papers coming out almost daily. As such, we will build a quick app to demonstrate an unfair coin. Question: Consider Two Coins, One Fair And One Unfair. But, could not arrive at a solution. This already is a pretty good estimate of the real bias! But you might want an even better estimate. The game is just like rolling a dice but with coins and tally marks. If we flip this coin three times, the sample space S is the following set of ordered triples: S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}. a) Calculate the theoretical probability of getting exactly 5 heads in 10 tosses when the probability of a head is 0. What is the probability that a fair coin lands Heads 6 times in a row? Ans: 1/26. Suppose that you're given a fair coin and you would like to simulate the probability distribution of repeatedly flipping a fair (six-sided) die. A fair coin gives you Heads. 322 Math Masters, p. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. In fact, the probability for most other values virtually disappeared — including the probability of the coin being fair (Bias = 0. The coin is flipped 50 times. But, could not arrive at a solution. Example question: What is the probability of rolling a 4 or 7 for two 6. A box contains 5 fair coins and 5 biased coins. Conditional Probability & the Rules of Probability; Intersection and Union of Sets (HSS-CP. However, it is not possible to bias a coin ﬂip—that. My initial idea is that we need to choose appropriate. In the fair coin experiment, there were 46 heads and 54 tails. But since there are 6 ways to get 2 heads, in four flips the probability of two heads is greater than that of any other result. For this problem we can use the normal approximation to the binomial distribution. An experiment in which a single action, such as flipping a coin, is repeated identically over and over. We can easily simulate an unfair coin by changing the probability p. In other words, the probability of getting 108 heads out of 200 coin tosses with a fair coin is 27%. Simulating a Biased Coin with a Fair Coin. share | cite | improve this question. (Or probability value) is a number between 0 and 1 inclusive associated with the likelihood of occurrence of a given event. The probability of having the unfair coin is $\frac13$. 5 Assuming an unfair coin (i. Probability is an estimate of the chance of winning divided by the total number of chances available. Now we must define the prior probability of seeing each of those values. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. In some situations, such as in flipping an unfair coin, we cannot calculate the theoretical probability. We can have either HTT, THT, or TTH. Image Transcriptionclose. the probability of success, p, is constant. After all, the probability that you will roll HHT right off the bat is , and the same for HTT and HHH, so there should be symmetry. When asked the question, what is the probability of a coin toss coming up heads, most people answer without hesitation that it is 50%, 1/2, or 0. The Probability Of Getting Heads On A Given Flip Of The Unfair Coin Is 0. You toss each coin 10 times (100 tosses in total) and observe results. Bernoulli Trials. The Binomial Likelihood Function Note the similarity between the probability function and the likelihood function; the right hand sides are the same. We flip a fair coin. Challenge the students to make an argument not based on the data as whether the game is Fair or Unfair and why. (Or probability value) is a number between 0 and 1 inclusive associated with the likelihood of occurrence of a given event. Conditional Probability & the Rules of Probability; Intersection and Union of Sets (HSS-CP. asked 23 hours ago. What is the probability that both children are girls? In other words, we want to find the probability that both children are girls, given that the family has at least one daughter named Lilia. Let R_p(r,n) be the probability that a run of r or more consecutive heads appears in n independent tosses of a coin (i. So, after 500 flips most of the probability gets distributed around the value 0. suppose the outcome heads occurs with probability 0. Increasing the repetitions, you can compare the paths taken in repea. help_outline. How many different ways can a reader choose 3 books out of 4, ignoring the order of selection? a. What is the probability of choosing two books of different colors? a. The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. In particular, it provides a "strategy" at the end to generate an unfair toss (which requires, e. You can change the weight or distribution of the coin by dragging the true probability bars (on the right in blue) up or down. Example question: What is the probability of rolling a 4 or 7 for two 6. The basic rule for probability is that you calculate it by looking at the number of possible outcomes in comparison to the outcome you’re interested in. I want it to start by having a dollar amount of x. Algebra -> Probability-and-statistics-> SOLUTION: An unfair coin has a probability 0. Up until now, we've looked at probabilities surrounding only equally likely events. 1 $\begingroup$ I am stuck on this question. e head or tail. Probability of switching coins = 0. Flipping coins is a classical way of thinking about probability. Additional ﬁgures show the probability distributions for n = 2,3,4,5,10. Coin toss probability Coin toss probability is explored here with simulation. Then a second coin is drawn at random from the box (without replacing the first one. If the coin is tossed 3 times, what is the probability that at least one of the tosses will turn up tails?. randomizing device is an unfair coin, with probability p ∈ (0,1) of heads. 6 and (P(tail)=. In this case, we can naturally assume that if the machines are unfair, the house will assign a lower probability to the expensive prizes, and higher probability to cheap ones. 6 that an "unfair" coin will turn up tails on any given toss. An unfair coin with 0. TOSSING A COIN M. Tutorials for Question - An unfair coin has a probability of coming up heads of 0. We express probability as a number between 0 and 1. An unfair coin has a probability of coming up heads of 0. Introduction The result of ntosses of a two-headed coin can be represented by. In simple terms, you have to figure out every possibility for what might happen. The probability of getting 3 lemons is 1/10 X 1/10 X 1/10, or 1/1000. Stating that it is "well known" that the probability of a coin landing on its side is 1/6000 is incorrect. It is not known whether a coin is fair or unfair. 2 is flipped. This is when the χ 2 test is important as it delineates whether 26:25 or 30:21 etc. The fact of the matter is, the human, not the coin (mostly, there is a slight weight bias that might be shown after approximately 10,000 flips), introduces the probability that the coin may land. An unfair coin is tossed; if a head appears uppermost, then a marble is selected from bag (X); otherwise, a marble is selected from bag (Y). probability of any continuous interval is given by p(a ≤ X ≤ b) = ∫f(x) dx =Area under f(X) from a to b b a That is, the probability of an interval is the same as the area cut off by that interval under the curve for the probability densities, when the random variable is continuous and the total area is equal to 1. Let t be the expected number of times you flip. Flipping the coin once is a Bernoulli trial, since there are exactly two complementary outcomes (flipping a head and flipping a tail), and they are both 1 2 \frac{1}{2} 2 1 no matter how many times the coin is flipped.