Chapter Five The Binomial Distribution and Related Topics
Section 2 Binomial Probabilities
Essential Question What are the three methods for solving binomial probability questions? Explain each of the three methods.
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.
Terms Bernoulli experiment Binomial experiment Continuous random variable Discrete random variable
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 0.00002349 0.0007480 0.1604 0.3400
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.
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 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?
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
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 = 1-0.70 = 0.30
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
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
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)!
Given n = 6, p = 0.1, find P(4): P( 4) C ( 0.1 ) 4 ( 6 4 0.9) 2 C ( 0.1 ) 4 ( 6 2 0.9) 2 6 5 2 1 0.1 30 2 0.1 15 0.0001 ( ) 4 ( 0.9) 2 ( ) 4 ( 0.9) 2 ( )( 0.81) 0.001215
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):
n = 8, p = 0.7, find P(6): P(6) 8 C 6 (.7)6 (.3) 2 8 C 2 (.7)6 (.3) 2 8 7 2 1 (.7)6 (.3) 2 56 2 (.7)6 (.3) 2 28(.1176)(.09) 0.2965
Table for Binomial Probability Appendix II Table 3 Pages A11 - A15
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): n r P = 0.70 8 : : : 4 0.1361 5 0.2541 6 0.2965 7 0.1977 8 0.0576
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 nine are cured.
Exactly four are cured: n = 12 r = 4 P(4) = 0.2311 p = 0.3 q = 0.7
n = 12 All are cured: r = 12 P(12) = 0.0000 p = 0.3 q = 0.7
None are cured: n = 12 r = 0 P(0) = 0.0138 p = 0.3 q = 0.7
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) 0.0015 + 0.0002 + 0.0000 + 0.0000 0.0017
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
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
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.
Find the Binomial Probability Suppose that the probability that a certain treatment cures a patient is 0.83. 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.
a. Exactly 8 are cured. n = 12 p = 0.83 r = 8 ( ) P r = 8 ( ) binompdf 12, 0.83, 8 0.09312
b. All twelve are cured. ( ) P r = 12 ( ) binompdf 12, 0.83, 12 0.1069
c. At most 6 are cured. ( ) P At most 6 ( ) P r 6 ( ) binomcdf 12, 0.83, 6 0.008752
d. At least nine are cured. ( ) P At least 9 P( r 9) 1 P( r 8) 1 binomcdf ( 12, 0.83, 8) 1 0.1324 0.8676
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) 0.6304 0.001460 0.6289
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