AP Statistics: Section 8.1A Binomial Probability

There are four conditions to a binomial setting: 1. Each observation falls into one of just two categories: _________ or ________. 2. There is a ______ number of observations, __. 3. These n observations are all ____________. 4. The probability of success, __, is _________ for each observation.

If data are produced in a binomial setting, then the random variable X = the number of successes is called a binomial random variable and the probability distribution of X is called a binomial distribution which is abbreviated ______.

Example 1: Determine if each of the following situations is a binomial setting. If so, state the probability distribution for X. If not, state which of the 4 conditions above is not met.

Situation 1: Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so having blood type O. A couple’s 5 different children inherit independently of each other. Let X = number of children with type O blood.

Situation 2: Deal 10 cards from a shuffled deck and let X = the number of red cards.

Situation 3: An engineer chooses a SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. Let X = the number of bad switches in the sample. ****When choosing an SRS from a population that is much larger than the sample, the observations are considered independent.

Binomial Probability: If X has a binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2, 3,..., n. If k is any one of these values, then

The notation is read n choose k and means the number of possible ways to choose k objects from a group of n objects. It is also written _____

Example 2 (combinations): How many different ways can we choose a subcommittee of size 3 from a student council that has 7 members?

Example 3: You randomly guess the answers to10 multiple choice questions which have 5 possible answers. What is the probability of getting exactly 6 correct answers?

Binomial Probability on the TI83/84:

Example 4: Consider situation 3 in example 1. Find the probability that in an SRS of size 10, no more than 1 switch fails.

Example 5: Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a big game, Corinne shoots 12 free throws and makes only 5 of them. Is it unusual for Corinne to perform this poorly? Note: We actually want the probability of making a basket on at most 5 free throws. Cumulative Binomial Probability on the TI83/84:

Any difference in answers between the manual calculation and using your calculator is due to rounding error.