Chapter Five. The Binomial Distribution and Related Topics

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1 Chapter Five The Binomial Distribution and Related Topics

2 Section 2 Binomial Probabilities

3 Essential Question What are the three methods for solving binomial probability questions? Explain each of the three methods.

4 Student Objectives The student will be able to identify the parts of a binomial experiment. The student will be able to compute the probability using the formula for a binomial experiment. The student will be able to compute the probability using the probability chart for a binomial experiment. The student will be able to compute the probability using a calculator for a binomial experiment. The student will be able to apply the binomial probability distributions to solve real-world problems.

5 Terms Bernoulli experiment Binomial experiment Continuous random variable Discrete random variable

6 Numerical Answers 1. Write down all the leading zeros after the decimal point until a non-zero digit is encountered. 2. Write the next three digits for a complete numerical answer. ACCEPTABLE ANSWERS

7 Features of a Binomial Experiment 1. There are a fixed number of trials. We denote this number by the letter n. 2. The n trials are independent and repeated under identical conditions.

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

9 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.

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

11 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

12 A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. Is this a binomial experiment?

13 Is this a binomial experiment? A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. Success = Failure = Hitting the target Not hitting the target

14 Is this a binomial experiment? A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. Probability of success = 0.70 Probability of failure = = 0.30

15 Is this a binomial experiment? A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. In this experiment there are n = trials. 8

16 Is this a binomial experiment? A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he 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 C n, r = n! r!(n r)!

18 Given n = 6, p = 0.1, find P(4): P( 4) C ( 0.1 ) 4 ( ) 2 C ( 0.1 ) 4 ( ) ( ) 4 ( 0.9) 2 ( ) 4 ( 0.9) 2 ( )( 0.81)

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

20 n = 8, p = 0.7, find P(6): P(6) 8 C 6 (.7)6 (.3) 2 8 C 2 (.7)6 (.3) (.7)6 (.3) (.7)6 (.3) 2 28(.1176)(.09)

21 Table for Binomial Probability Appendix II Table 3 Pages A11 - A15

22 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.

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

24 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 nine are cured.

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

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

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

28 At least six are cured: n = 12 r 9 p = 0.3 q = 0.7 P( At least 9) P( r = 9, r = 10, r = 11, or r = 12) P( r = 9) + P( r = 10) + P( r = 11) + P( r = 12)

29 Using your calculator to determine Binomial Probability Single r value 2 nd DISTR DISTR Option A: binompdf Range of r values 2 nd DISTR DISTR Option B: binomcdf

30 For a single r value Single r value 2 nd DISTR DISTR Option A: binompdf ( ) binompdf n, p, r binompdf Number of trials, Probability of a success, Number of successes

31 Range of r values Range of r values 2 nd DISTR DISTR Option B: binomcdf binomcdf binomcdf n, p, r Number of trials *** WARNING ***, Probability of a success ( ), Number of successes from 0 up to an including this value This calculuation will calculate the sum of ALL the probabilities from r = 0 up to and including the given r value.

32 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 8 are cured. b. All twelve are cured. c. At most 6 are cured. d. At least nine are cured. e. At least six but not more than 10 are cured.

33 a. Exactly 8 are cured. n = 12 p = 0.83 r = 8 ( ) P r = 8 ( ) binompdf 12, 0.83,

34 b. All twelve are cured. ( ) P r = 12 ( ) binompdf 12, 0.83,

35 c. At most 6 are cured. ( ) P At most 6 ( ) P r 6 ( ) binomcdf 12, 0.83,

36 d. At least nine are cured. ( ) P At least 9 P( r 9) 1 P( r 8) 1 binomcdf ( 12, 0.83, 8)

37 e. At least six but not more than 10 are cured. P( At least 6 but not more than 10) P( 6 r 10) P( r 10) P( r 5) binomcdf ( 12, 0.83, 10) binomcdf ( 12, 0.83, 5)

Binomial Probability

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