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 P(r) C n, r p r q n r where C n, r binomial coefficient r!(n n! r)!
Calculating Binomial Probability Given n = 6, p = 0.1, find P(4): P(4) 6! (.1) 4 (.9) 2 4!(6 4)! 15(.0001)(.81) 0.001215
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. P(6) n = 8, p = 0.7, find P(6): 8! (.7) 6!2! 28(.117649)(.09) 6 (.3) 2 0.2964755
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): p.70 n r 8. 4 5.254 6 7.296.198
Find the Binomial Probability Suppose that the probability that a certain treatment cures a patient is 0.30. 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 P(4) = 0.231 q = 0.7
All are cured: n = 12 r = 12 p = 0.3 P(12) = 0.000 q = 0.7
None are cured: n = 12 r = 0 p = 0.3 P(0) = 0.014 q = 0.7
r =? At least six are cured:
At least six are cured: r = 6, 7, 8, 9, 10, 11, or 12 P(6) =.079 P(7) =.029 P(8) =.008 P(10) =.000 P(11) =.000 P(12) =.000 P(9) =.001
At least six are cured: P( 6, 7, 8, 9, 10, 11, or 12) =.079 +.029 +.008 +.001 +.000 + =.117.000 +.000