Binomial Probability

Features of a Binomial Experiment 1. There are a fixed number of trials. We denote this number by the letter n.

Features of a Binomial Experiment 2. The n trials are independent and repeated under identical conditions.

Features of a Binomial Experiment 3. Each trial has only two outcomes: success, denoted by S, and failure, denoted by F.

Features of a Binomial Experiment 4. For each individual trial, the probability of success is the same. We denote the probability of success by p and the probability of failure by q. Since each trial results in either success or failure, p + q = 1 and q = 1 – p.

Features of a Binomial Experiment 5. The central problem is to find the probability of r successes out of n trials.

Binomial Experiments Repeated, independent trials Number of trials = n Two outcomes per trial: success (S) and failure (F) Number of successes = r Probability of success = p Probability of failure = q = 1 – p

A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Is this a binomial experiment?

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = failure =

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = hitting the target failure = not hitting the target

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = Probability of failure =

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = 0.70 Probability of failure = 1 – 0.70 = 0.30

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _____ trials.

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _8__ trials.

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = _____.

Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = _ 6__.

Binomial Probability Formula

Calculating Binomial Probability Given n = 6, p = 0.1, find P(4):

Calculating Binomial Probability A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. n = 8, p = 0.7, find P(6):

Table for Binomial Probability Table 3 Appendix II

Using the Binomial Probability Table Find the section labeled with your value of n. Find the entry in the column headed with your value of p and row labeled with the r value of interest.

Using the Binomial Probability Table n = 8, p = 0.7, find P(6):

Find the Binomial Probability Suppose that the probability that a certain treatment cures a patient is Twelve randomly selected patients are given the treatment. Find the probability that: a.exactly 4 are cured. b.all twelve are cured. c.none are cured. d.at least six are cured.

Exactly four are cured: n = r = p = q =

Exactly four are cured: n = 12 r = 4 p = 0.3 q = 0.7 P(4) = 0.231

All are cured: n = 12 r = 12 p = 0.3 q = 0.7 P(12) = 0.000

None are cured: n = 12 r = 0 p = 0.3 q = 0.7 P(0) = 0.014

At least six are cured: r = ?

At least six are cured: r = 6, 7, 8, 9, 10, 11, or 12 P(6) =.079 P(7) =.029 P(8) =.008 P(9) =.001 P(10) =.000 P(11) =.000 P(12) =.000

At least six are cured: P( 6, 7, 8, 9, 10, 11, or 12) = =.117