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Binomial Distributions for AP Statistics

By — McGraw-Hill Professional
Updated on Feb 4, 2011

Practice problems for these concepts can be found at:

A binomial experiment has the following properties:

  • The experiment consists of a fixed number, n, of identical trials.
  • There are only two possible outcomes (that's the "bi" in "binomial"): success (S) or failure (F).
  • The probability of success, p, is the same for each trial.
  • The trials are independent (that is, knowledge of the outcomes of earlier trials does not affect the probability of success of the next trial).
  • Our interest is in a binomial random variable X, which is the count of successes in n trials. The probability distribution of X is the binomial distribution.

(Taken together, the second, third, and fourth bullets above are called Bernoulli trials. One way to think of a binomial setting is as a fixed number n of Bernoulli trials in which our random variable of interest is the count of successes X in the n trials. You do not need to know the term Bernoulli trials for the AP exam.)

The short version of this is to say that a binomial experiment consists of n independent trials of an experiment that has two possible outcomes (success or failure), each trial having the same probability of success (p). The binomial random variable X is the count of successes.

In practice, we may consider a situation to be binomial when, in fact, the independence condition is not quite satisfied. This occurs when the probability of occurrence of a given trial is affected only slightly by prior trials. For example, suppose that the probability of a defect in a manufacturing process is 0.0005. That is, there is, on average, only 1 defect in 2000 items. Suppose we check a sample of 10,000 items for defects. When we check the first item, the proportion of defects remaining changes slightly for the remaining 9,999 items in the sample. We would expect 5 out of 10,000 (0.0005) to be defective. But if the first one we look at is not defective, the probability of the next one being defective has changed to 5/9999 or 0.0005005. It's a small change but it means that the trials are not, strictly speaking, independent. A common rule of thumb is that we will consider a situation to be binomial if the population size is at least 10 times the sample size.

Symbolically, for the binomial random variable X, we say X has B(n, p).

example: Suppose Dolores is a 65% free throw shooter. If we assume that that repeated shots are independent, we could ask, "What is the probability that Dolores makes exactly 7 of her next 10 free throws?" If X is the binomial random variable that gives us the count of successes for this experiment, then we say that X has B(10,0.65). Our question is then: P(X = 7) = ?.

We can think of B(n,p,x) as a particular binomial probability. In this example, then, B(10,0.65,7) is the probability that there are exactly 7 successes in 10 repetitions of a binomial experiment where p = 0.65. This is handy because it is the same syntax used by the TI-83/84 calculator (binompdf(n,p,x)) when doing binomial problems.

If X has B(n,p), then X can take on the values 0,1,2,…, n. Then,

gives the binomial probability of exactly x successes for a binomial random variable X that has B(n, p).

Now,

Binomial Distributions

On the TI-83/84,

Binomial Distributions

and this is found in the MATH PRB menu. n! ("n factorial") means n(n – 1)(n – 2)… (2)(1,), and the factorial symbol can be found in the MATH PRB menu.

example: Find B(15,.3,5). That is, find P(X = 5) for a 15 trials of a binomial random variable X that succeeds with probability 3.

solution:

Binomial Distributions

(On the TI-83/84, = nCCrr can be found in the MATH PRB menu. To get , enter 15nCr5.)

Binomial Distributions

example: Consider once again our free-throw shooter (Dolores) from an earlier example. Dolores is a 65% free-throw shooter and each shot is independent. If X is the count of free throws made by Dolores, then X has B(10, 0.65) if she shoots 10 free throws. What is P(X = 7)?

solution:

Binomial Distributions

example: What is the probability that Dolores makes no more than 5 free throws? That is, what is P(X ≤ 5)?

solution:

Binomial Distributions

There is about a 25% chance that she will make 5 or fewer free throws. The solution to this problem using the calculator is given by binomcdf (10,0.65,5).

example: What is the probability that Dolores makes at least 6 free throws?

solution: P(X ≥ 6) = P(X = 6) + P(X = 7) +… + P(X = 10) = 1-binomcdf(10,0.65,5)=0.751.

(Note that P(X > 6) = 1 – binomcdf(10,0.65,6)).

The mean and standard deviation of a binomial random variable X are given by . A binomial distribution for a given n and p (meaning you have all possible values of x along with their corresponding probabilities) is an example of a probability distribution as defined in Chapter 7. The mean and standard deviation of a binomial random variable X could be found by using the formulas from Chapter 7,

but clearly the formulas for the binomial are easier to use. Be careful that you don't try to use the formulas for the mean and standard deviation of a binomial random variable for a discrete random variable that is not binomial.

    example: Find the mean and standard deviation of a binomial random variable X that has B(85, 0.6).
    Binomial Distributions

Practice problems for these concepts can be found at:

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