4.1 Binomial DistributionDay 1

There are many experiments in which the results of each trial can be reduced to 2 outcomes.

Binomial Experiment:There are n independent trials.Each trial has only 2 possible outcomes:

Success or failure

The probability of success is the same for each trial.

The probability of success is p. The probability of failure is 1-p.

Finding a Binomial Probability

For a binomial experiment consisting of n trials, the probability of exactly k successes is:

P(k successes) = nCk pk (1-p)n-k

where the probability of success on each trial is p.

Example

According to a survey taken by USA Today, about 37% of adults believe that Unidentified Flying Objects (UFOs) really exist. Suppose you randomly survey 6 adults. What is the probability that exactly 2 of them believe that UFOs exist?

SolutionLet p = 0.37

Survey 6 adults; n = 6

k = 2

P(k = 2) = 6C2(0.37)2(1- 0.37)6 – 2

= 0.323

The probability that exactly 2 of the people surveyed believe that UFOs really exist is about 32%.

42 )63.0()37.0(!2!4

!6

You Try

At a college, 53% of students receive financial aid. In a random group of 9 students, what is the probability that exactly 5 of them receive financial aid?

p=.53 (the probability of success for each trial) n=9 (number of different trials or experiments) k=5 (the probability of getting 5 successes)

P(k=5) = 9C5 .535 (1-.53)9-5

≈ 26%

EXAMPLE Draw a histogram of the binomial

distribution for the survey from the first Example of UFOs. Then find the probability that at most 2 of the people surveyed believe that UFOs really exist.

Hint: Use P(k successes) = nCk pk (1-p)n-k

Solution

P(k = 0) = 6C0(0.37)0(0.63)6 ≈ 0.063

P(k = 1) = 6C1(0.37)1(0.63)5 ≈ 0.220

P(k = 2) = 6C2(0.37)2(0.63)4 ≈ 0.323

P(k = 3) = 6C3(0.37)3(0.63)3 ≈ 0.253

P(k = 4) = 6C4(0.37)4(0.63)2 ≈ 0.112

P(k = 5) = 6C5(0.37)5(0.63)1 ≈ 0.026

P(k = 6) = 6C6(0.37)6(0.63)0 ≈ 0.003

Solution Con’t

Solution Con’t

The probability of getting at most k = 2 successes is

P(k < 2) = P(2) + P(1) + P(0)

= 0.323 + 0.220 + 0.063

= 0.606

The probability that at most 2 of the people surveyed believe that UFOs really exist is about 61%

You Try

Draw a histogram of the binomial distribution for the class of students from the previous You Try .

Hint: Use P(k successes) = nCk pk (1-p)n-k

P(k=0) = 9C0 .530 (1-.53)9-0 = .001P(k=1) = 9C1 .531 (1-.53)9-1 = .011P(k=2) = 9C2 .532 (1-.53)9-2 = .05P(k=3) = 9C3 .533 (1-.53)9-3 = .13P(k=4) = 9C4 .534 (1-.53)9-4 = .23P(k=5) = 9C5 .535 (1-.53)9-5 = .26P(k=6) = 9C6 .536 (1-.53)9-6 = .19P(k=7) = 9C7 .537 (1-.53)9-7 = .09P(k=8) = 9C8 .538 (1-.53)9-8 = .03P(k=9) = 9C9 .539 (1-.53)9-9 = .003

EXAMPLE

Find the probability that fewer than 3 students in the class receive financial aid.

= P(0) + P(1) + P(2) = .001 + .011 + .05= 0.062

EXAMPLE