12.6 Binomial Distributions 2011
February 07, 2011
12.6 Binomial Distributions
Objective:
• Find binomial probabilities.
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12.6 Binomial Distributions 2011
February 07, 2011
Warm­up
In problems 1 ­ 3, evaluate each expression. PRACTICE FOR THE TEST & SHOW YOUR WORK!!
1. 4C2
2. 3C3
3. 6C0
In problems 4 ­ 5, expand each binomial.
4. (w ­ y)4
5. (a + 2b)5
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12.6 Binomial Distributions 2011
February 07, 2011
Binomial Experiment
A binomial experiment has three important features:
• The situation involves repeated trials.
• Each trial has two possible outcomes (success or failure).
• The probability of success is constant throughout the trials. 3
12.6 Binomial Distributions 2011
February 07, 2011
Designing a Binomial Experiment
Suppose you guess the answers to three questions of a multiple­choice test. Each question has five choices, with one correct choice.
1. In this situation, describe a trial.
Each guess is a trial.
2. How many trials are there in this situation?
There are 3 trials.
3. In this situation, describe a success.
Guessing a correct answer is a success.
4. What is the probability of success on any single trial?
The probability of success is 1 out of 5 so we will use 0.2.
5. What is the probability of failure on any single trial?
The probability of failure is 4 out of 5 so we will use 0.8.
Note: success + failure = 1 0.2 + 0.8 = 1
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12.6 Binomial Distributions 2011
February 07, 2011
Designing a Binomial Experiment
Suppose you guess the answers to three questions of a multiple­choice test. Each question has five choices, with one correct choice.
Binomial Probability:
In the above situation, what is the probability that you will guess the correct answer for exactly 2 of the 3 questions?
(0.2)(0.2)(0.8) + (0.2)(0.8)(0.2) + (0.8)(0.2)(0.2) + 0.032 + 0.032
0.032
0.096
We use combination because we are not looking for order repeats.
2
1
3 2
C (0.2) (0.8) = 0.096
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12.6 Binomial Distributions 2011
February 07, 2011
Binomial Probability
x
nCx(p)
(n ­ x)
(q)
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12.6 Binomial Distributions 2011
February 07, 2011
Example #1: As part of a promotion, a store is giving away scratch­off cards with in­store discounts. Prizes are awarded on 35% of the game cards. Suppose you have five cards. Find the probability that exactly three of the five cards will reveal a prize.
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12.6 Binomial Distributions 2011
February 07, 2011
Example #2: Assume that Chauncey's probability for success on any free throw is the same as his cumulative record to date (89.3%). Find the probability that he will make exactly 6 out of 10 consecutive free throws. P(6 out of 10) = 10C6 p6q4
= 10C6 (0.893)6(0.107)4
= 210 (0.893)6(0.107)4
= 0.01395928 ≈ 1.4% chance
Chauncey Billups
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12.6 Binomial Distributions 2011
February 07, 2011
Example #3: A scientist hopes to launch a weather balloon on one of the next three mornings. For each morning, there is a 40% chance of suitable weather. What is the probability that there will be at least one morning with suitable weather?
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12.6 Binomial Distributions 2011
February 07, 2011
Example #4: A calculator contains four batteries. With normal use, each battery has 92% chance of lasting for one year. What is the probability that all four batteries will last a year? What is the probability that 2 or less batteries will last a year?
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12.6 Binomial Distributions 2011
February 07, 2011
Example #5: One survey found that 80% of respondents eat corn on the cob in circles rather that from side to side. Assume that this sample accurately represents the population. What is the probability that, out of five people you know, at least two of them eat corn on the cob in circles?
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12.6 Binomial Distributions 2011
February 07, 2011
Homework
page 689
(15 ­ 16, 18 ­ 20, 24, 26)
&
page 691 (#8)
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