Thursday, October 22

# Fair coin distribution

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Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. Of course 0. But there's something wrong. Shouldn't the sum of probabilities give 1?

Except from the two extremes, i. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Probability distribution, unfair coin Ask Question. Asked 5 years, 6 months ago.

The probability distribution A list of each possible value and its probability. The probabilities in the probability distribution of a random variable X must satisfy the following two conditions:. A fair coin is tossed twice. Let X be the number of heads that are observed. The possible values that X can take are 0, 1, and 2. The probability of each of these events, hence of the corresponding value of Xcan be found simply by counting, to give.

This table is the probability distribution of X. A histogram that graphically illustrates the probability distribution is given in Figure 4. Figure 4. A pair of fair dice is rolled.

Let X denote the sum of the number of dots on the top faces. The possible values for X are the numbers 2 through Continuing this way we obtain the table. Before we immediately jump to the conclusion that the probability that X takes an even value must be 0. We compute. The mean of a random variable may be interpreted as the average of the values assumed by the random variable in repeated trials of the experiment.

Find the mean of the discrete random variable X whose probability distribution is. A service organization in a large town organizes a raffle each month. Each has an equal chance of winning. Let X denote the net gain from the purchase of one ticket. Let W denote the event that a ticket is selected to win one of the prizes.

Using the table. The negative value means that one loses money on the average. In particular, if someone were to buy tickets repeatedly, then although he would win now and then, on average he would lose 40 cents per ticket purchased.After studying the random variables and discrete probability distributions, we need to look a little more closely at a special type of discrete distribution, one that is closely related to the example we used earlier about the number of boys when you have 4 kids.

But the mathematics is really the same. Suppose we have a fair coin so the heads-on probability is 0. If we let the random variable X represent the number of heads in the 3 tosses, then clearly, X is a discrete random variable, and can take values ranging from 0 to 3. Moreover, we can represent the probability distribution of X in the following table:. What will be the probability of? So the chance of 0 heads is just the probability of getting all tails, i. Alternatively, we can also use independence of the 3 coin tosses, and break down the probability using the multiplication rule, applied to the independent events:.

Using the same reasoning, it's also easy to determine that. So our probability distribution now becomes:. So the probability of getting 1 head is. But there is another way that uses the additional rule and multiplication rule together that you should know: since "1 head" is technically a union of the three outcomes: HTT, THT, TTH, and they are mutually exclusive; we may use the addition rule applied to the mutually exclusive events:.

You probably noticed that the three probabilities being added are all the same, so we can shorten it as:. Using the same type of analysis, we can also determine that.

Hence the completed probability distribution is the following:. The "at least" phrasing means we're looking for the probability of the coin landing heads two or more times. The "at most" phrasing means we're looking for the probability of the coin landing tails two times or fewer.

What you just saw was a binomial distributionwhich is the generalized version of a fixed number of coin flips. Here are the assumptions of the binomial distribution that were listed in the lecture:. For the experiment above, the number of trailsand the probability of success. Using these two parameters, we can determine the entire probability distribution.

To see the flexibility of the binomial distribution, let's imagine that someone glued some chewing gum on one side of the coin on a side note, one of my previous Math 15 students did this as part of his term project. So we know this can be done. As a result, the coin is no longer fair.Binomial distribution is a discrete probability distribution when an experiment with two possible sets of an outcomes is carried out.

The trials are done with replacement and probability of success remains the same for all trials. The outcomes are also independent from the preceding result of a trial. The number of heads success in 11 number of tosses follows binomial distribution with following parameters:. Try it risk-free for 30 days.

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Question: A fair coin is tossed 11 times. What is the probability that at least two heads appear? Binomial Distribution: Binomial distribution is a discrete probability distribution when an experiment with two possible sets of an outcomes is carried out. Learn more about this topic:. Try it risk-free. Introduction to Statistics Textbook. Introduction to Statistics: Certificate Program. Intro to Criminal Justice: Help and Review. Intro to Anthropology: Help and Review. American Government: Help and Review. Business Organizational Behavior. Introduction to Political Science: Help and Review.

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Earth Science Weather and Climate. Education Foundations of Education. Social Psychology: Tutoring Solution. History The Civil War and Reconstruction.In statisticsthe question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory.

The practical problem of checking whether a coin is fair might be considered as easily solved by performing a sufficiently large number of trials, but statistics and probability theory can provide guidance on two types of question; specifically those of how many trials to undertake and of the accuracy an estimate of the probability of turning up heads, derived from a given sample of trials.

A fair coin is an idealized randomizing device with two states usually named "heads" and "tails" which are equally likely to occur.

It is based on the coin flip used widely in sports and other situations where it is required to give two parties the same chance of winning.

Either a specially designed chip or more usually a simple currency coin is used, although the latter might be slightly "unfair" due to an asymmetrical weight distribution, which might cause one state to occur more frequently than the other, giving one party an unfair advantage.

It is of course impossible to rule out arbitrarily small deviations from fairness such as might be expected to affect only one flip in a lifetime of flipping; also it is always possible for an unfair or " biased " coin to happen to turn up exactly 10 heads in 20 flips. Therefore, any fairness test must only establish a certain degree of confidence in a certain degree of fairness a certain maximum bias. In more rigorous terminology, the problem is of determining the parameters of a Bernoulli processgiven only a limited sample of Bernoulli trials.

This article describes experimental procedures for determining whether a coin is fair or unfair. There are many statistical methods for analyzing such an experimental procedure. This article illustrates two of them. Both methods prescribe an experiment or trial in which the coin is tossed many times and the result of each toss is recorded.

The results can then be analysed statistically to decide whether the coin is "fair" or "probably not fair". An important difference between these two approaches is that the first approach gives some weight to one's prior experience of tossing coins, while the second does not.

The question of how much weight to give to prior experience, depending on the quality credibility of that experience, is discussed under credibility theory. One method is to calculate the posterior probability density function of Bayesian probability theory.

A test is performed by tossing the coin N times and noting the observed numbers of heads, hand tails, t. The symbols H and T represent more generalised variables expressing the numbers of heads and tails respectively that might have been observed in the experiment.

Next, let r be the actual probability of obtaining heads in a single toss of the coin. This is the property of the coin which is being investigated. Using Bayes' theoremthe posterior probability density of r conditional on h and t is expressed as follows:. The prior probability density distribution summarizes what is known about the distribution of r in the absence of any observation.Understanding probability.

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Binomial Distribution Probablity Coins

Tough Algebra Word Problems. If you can solve these problems with no help, you must be a genius! Real Life Math Skills Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. All right reserved. Homepage Free math problems solver!When you flip a coin, there are two possible outcomes: heads and tails.

Each outcome has a fixed probability, the same from trial to trial. More generally, there are situations in which the coin is biased, so that heads and tails have different probabilities. In the present section, we consider probability distributions for which there are just two possible outcomes with fixed probabilities summing to one. These distributions are called binomial distributions. The four possible outcomes that could occur if you flipped a coin twice are listed below in Table 1.

To see this, note that the tosses of the coin are independent neither affects the other. The same calculation applies to the probability of a head on Flip 1 and a tail on Flip 2. The four possible outcomes can be classified in terms of the number of heads that come up. The number could be two Outcome 1one Outcomes 2 and 3 or 0 Outcome 4. The probabilities of these possibilities are shown in Table 2 and in Figure 1.

Table 2 summarizes the situation. Figure 1 is a discrete probability distribution: It shows the probability for each of the values on the X-axis. Defining a head as a "success," Figure 1 shows the probability of 0, 1, and 2 successes for two trials flips for an event that has a probability of 0.

This makes Figure 1 an example of a binomial distribution. The formula for the binomial distribution is shown below:. Applying this to the coin flip example. If you flip a coin twice, what is the probability of getting one or more heads?

Since the probability of getting exactly one head is 0. Now suppose that the coin is biased. The probability of heads is only 0. What is the probability of getting heads at least once in two tosses? Substituting into the general formula above, you should obtain the answer.

We toss a coin 12 times. What is the probability that we get from 0 to 3 heads? The answer is found by computing the probability of exactly 0 heads, exactly 1 head, exactly 2 heads, and exactly 3 heads. The probability of getting from 0 to 3 heads is then the sum of these probabilities.

The probabilities are: 0. The sum of the probabilities is 0. The calculation of cumulative binomial probabilities can be quite tedious. Therefore we have provided a binomial calculator to make it easy to calculate these probabilities. Binomial Calculator. Consider a coin-tossing experiment in which you tossed a coin 12 times and recorded the number of heads.

If you performed this experiment over and over again, what would the mean number of heads be? On average, you would expect half the coin tosses to come up heads.