Binomial Probability Distribution: Understanding Binomial Experiments and Probabilities - , Exams of Statistics

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The Binomial Probability Distribution

MATH 130, Elements of Statistics I

J. Robert Buchanan

Department of Mathematics

Spring 2008

Binomial Experiments

A binomial experiment repeats a simple experiment several times. The simple experiment has only two outcomes. The binomial experiment counts the number of outcomes of each of the two types.

Example Flip a coin 10 times and count the number of heads and tails that occur.

Notation

Let n be the number of independent trials of the experiment. Let p be the probability of success (and 1 − p be the probability of failure). Let X be the random variable denoting the number of successes in the n trials of the binomial experiment.

0 ≤ X ≤ n

Examples

Example Which if the following situations describe binomial experiments? (^1) A test consists of 10 True/False questions and X represents the number of questions answered correctly by guessing. (^2) A test consists of 10 multiple choice (5 choices per question) questions and X represents the number of questions answered correctly by guessing. (^3) An experiment consists of drawing five cards from a well-shuffled deck with replacement. The drawn card is identified as a “heart” or “not a heart”. Random variable X represents the number of hearts drawn. (^4) An experiment consists of drawing five cards from a well-shuffled deck without replacement. The drawn card is identified as a “heart” or “not a heart”. Random variable X

Binomial Probabilities

The probability of x successes out of n trials of a binomial experiment for which the probability of success on a single trial is p is P ( x ) = (^) nCx px^ ( 1 − p ) n − x^ , for x = 0 , 1 ,... , n.

Example What is the probability that in 12 flips of a fair coin that exactly 4 heads will result?

Binomial Probability Tables

Table II of Appendix A (pages A–2 through A–5) lists pre-computed values of the binomial probability formula. Table II summarizes the cases of n = 2 , 3 ,... , 12 , 15 , 20. The binomial probabilities for p = 0. 01 , 0. 05 , 0. 10 ,... , 0. 95 are listed.

Example What is the probability that in 12 flips of a fair coin that exactly 7 heads will result?

Binomial Probability Tables (cont.)

Table III of Appendix A (pages A–6 through A–9) lists pre-computed cumulative values of the binomial probability formula. The cumulative value is P ( x ≤ m ),

P ( x ≤ m ) =

∑^ m

i = 0

nCi pi^ (^1 −^ p ) n − i

Table III summarizes the cases of n = 2 , 3 ,... , 12 , 15 , 20. The binomial probabilities for p = 0. 01 , 0. 05 , 0. 10 ,... , 0. 95 are listed.

Example What is the probability that in 12 flips of a fair coin that 7 or fewer heads will result?

Binomial Probability Tables (cont.)

Table III of Appendix A (pages A–6 through A–9) lists pre-computed cumulative values of the binomial probability formula. The cumulative value is P ( x ≤ m ),

P ( x ≤ m ) =

∑^ m

i = 0

nCi pi^ (^1 −^ p ) n − i

Table III summarizes the cases of n = 2 , 3 ,... , 12 , 15 , 20. The binomial probabilities for p = 0. 01 , 0. 05 , 0. 10 ,... , 0. 95 are listed.

Example What is the probability that in 12 flips of a fair coin that 7 or fewer heads will result?

Mean and Standard Deviation

Theorem A binomial experiment with n independent trials and probability of success p on a trial has a mean and standard deviation given by the formulas:

μ X = np σ X =

np ( 1 − p ).

Examples

Example There is a 90% chance that a pizza from TelePizza will be delivered in less that 30 minutes. If a pizza is not delivered in less than 30 minutes, the next order is free. Suppose TelePizza must process 300 delivery orders per day. What is the mean and standard deviation in the number of pizzas delivered on time?

Observation

As the number of trials n of a binomial experiment increases, the probability distribution of the random variable X becomes bell-shaped. If np ( 1 − p ) ≥ 10, the probability distribution will be bell-shaped. Hence when np ( 1 − p ) ≥ 10 we may use the Empirical Rule to identify unusual observations in a binomial experiment.

Example

Example There is a 90% chance that a pizza from TelePizza will be delivered in less that 30 minutes. If a pizza is not delivered in less than 30 minutes, the next order is free. Suppose TelePizza must process 300 delivery orders per day. (^1) According to the Empirical Rule, between what two values would 95% of the daily on-time deliveries fall? (^2) Would it be unusual to find that only 244 pizzas out of 300 were delivered on time?

Homework

Read Section 6.2. Pages 310-312: 7–33 odd, 35, 39, 43, 47