Binomial Probability
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1 Binomial Probability
2 Features of a Binomial Experiment 1. There are a fixed number of trials. We denote this number by the letter n.
3 Features of a Binomial Experiment 2. The n trials are independent and repeated under identical conditions.
4 Features of a Binomial Experiment 3. Each trial has only two outcomes: success, denoted by S, and failure, denoted by F.
5 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.
6 Features of a Binomial Experiment 5. The central problem is to find the probability of r successes out of n trials.
7 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
8 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?
9 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 =
10 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
11 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 =
12 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 = = 0.30
13 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.
14 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.
15 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 =.
16 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.
17 Binomial Probability Formula P(r) C n, r p r q n r where C n, r binomial coefficient r!(n n! r)!
18 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)
19 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( )(.09) 6 (.3)
20 Table for Binomial Probability Table 3 Appendix II
21 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.
22 Using the Binomial Probability Table n = 8, p = 0.7, find P(6): p.70 n r
23 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.
24 Exactly four are cured: n = r = p = q =
25 Exactly four are cured: n = 12 r = 4 p = 0.3 P(4) = q = 0.7
26 All are cured: n = 12 r = 12 p = 0.3 P(12) = q = 0.7
27 None are cured: n = 12 r = 0 p = 0.3 P(0) = q = 0.7
28 r =? At least six are cured:
29 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
30 At least six are cured: P( 6, 7, 8, 9, 10, 11, or 12) = =
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