Download AP Stats - Aim 40 - Manhasset Public Schools

AP Statistics ­ Aim #40 completed.notebook
Unit 3 - Chapter 8
Binomial and Geometric Distributions
Aim #40 - How do we determine a binomial distribution?
Binomial Setting:
1. Each observation is either a "success" or "failure"
2. There is a fixed number n of observations
3. The n observations are all independent
4. The probability of success, p, is the same for each observation.
Examples
flipping a coin
shooting a free throw
number of girls in a family
defective lightbulbs in a package of 100
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AP Statistics ­ Aim #40 completed.notebook
Buzz Words:
binomial distribution - the distribution of the count X of successes
in the binomial setting
Buzz Symbols:
n - number of observations
p - probability of success
X is B(n, p)
X - whole numbers from 0 to n
eg. Are these examples of a binomial setting? Why or why not?
a) An auto manufacturer chooses one car from each hour's
production for a detailed quality inspection. One variable
recorded is the count X of finish defects in the car's paint.
b) The pool of potential jurors for a murder case contains 100
persons chosen at random from the adult residents of a large
city. Each person in the pool is asked whether he or she
opposes the death penalty; X is the number who say "yes."
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AP Statistics ­ Aim #40 completed.notebook
c) Joe buys a ticket in his state's "Pick 3" lottery game every
week; X is the number of times in the year that he wins a prize.
k
n-k
P(X = k) = nCk p (1-p)
eg. A waiter knows from experience that 7 out of 10 people
who dine alone will leave a tip. Tuesday evening, he served 12
lone diners. What is the probability that the waiter received a
tip from exactly 9 of these diners?
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AP Statistics ­ Aim #40 completed.notebook
eg. What is the probability that a 2 shows on three of the dice
when four dice are tossed?
eg. A company that makes breakfast cereal puts a coupon for a
free box of cereal in 3 out of every 20 boxes. What is the
probability that Mrs. Sullivan will find 2 coupons in the next 5
boxes of cereal that she buys?
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AP Statistics ­ Aim #40 completed.notebook
More Buzz Words:
probability distribution function (pdf): Given a discrete random
variable X, the pdf assigns a probability to each value of X
eg. Complete the pdf for the same problem: A company that
makes breakfast cereal puts a coupon for a free box of cereal
in 3 out of every 20 boxes. What is the probability that Mrs.
Sullivan will find 0, 1, 2, 3, 4 or 5 coupons in the next 5 boxes of
cereal that she buys? Use the calculator!
X:
Prob:
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AP Statistics ­ Aim #40 completed.notebook
More Buzz Words (cont):
cumulative distribution function (cdf): Given a random variable
X, the cdf of X calculates the sum of the probabilities for 0, 1,
2, . . . up to the value X. That is, it calculates the probability of
obtaining AT MOST X successes in n trials.
binomialcdf(n, p, X)
How would you determine AT LEAST?
eg. Same problem: What is the probability that Mrs. Sullivan
will find AT MOST 2 coupons in the next 5 boxes of cereal that
she buys? Use the calculator!
What is the probability that she will find AT LEAST 4 coupons?
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AP Statistics ­ Aim #40 completed.notebook
eg. Each child born to a particular set of parents has
probability of 0.25 of having blood type O. Suppose these
parents have 5 children. Let X = the number of children who
have blood type O.
a) What is the probability that exactly 2 of the children have
type O blood? Write out calculator notation and binomial
distribution notation.
26.4%
b) What is the probability that at most 4 children have type O
blood? Write out calculator notation and binomial distribution
notation using sigma.
99.9%
c) What is the probability that at least 3 children have type O
blood?
10.4%
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AP Statistics ­ Aim #40 completed.notebook
eg. Suppose that Doug guesses on each question of a 50-item
true/false quiz. Find the probability that Doug passes if:
a) A score of 25 or more is needed to pass
55.6%
b) A score of 32 or more is needed to pass
3.4%
Wrap it Up!
Laura guesses on each question of a multiple choice quiz. If
each question has 4 different choices, find the probability that
Laura gets one or more correct answers on a 10-item quiz.
94.4%
Assignment #40 - Worksheet #40
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