Download AP Statistics – Ch 8 – The Binomial and Geometric Distributions

AP Statistics – Ch 8 – The Binomial and Geometric Distributions
Ch 8.1 – The Binomial Distributions
The Binomial Setting
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A situation where these four conditions are satisfied is called a binomial setting.
1. Each observation falls into one of just two categories, which we call “success”
or “failure”.
2. There is a fixed number of n observations.
3. The n observations are all independent.
4. The probability of success, call it p, is the same for each observation.
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If data are produced in a binomial setting, then the random variable X = # of
successes is called a binomial random variable. The probability distribution of X
is called a binomial distribution.
The distribution of the count X of successes in the binomial setting is the binomial
distribution with parameters n and p. Abbreviated, we say X is B(n, p).
The parameter n is the number of observations, and p is the probability of a
success on any one observation.
The possible values of X are the whole numbers from 0 to n.
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Example – Binomial Setting
o Indicate whether a binomial distribution is a reasonable model for the
random variable X. Give your reasons in each case.
o A manufacturer produces a large number of toasters. From past
experience, the manufacturer knows that approximately 2% are defective.
In a quality control procedure, we randomly select 20 toasters for testing.
We want to determine the probability that no more than one of these
toasters is defective.
o Draw a card from a standard deck of 52 playing cards, observe the card,
and replace the card within the deck. Count the number of times you draw
a card in this manner until you observe a jack.
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AP Statistics – Ch 8 – The Binomial and Geometric Distributions
Finding Binomial Probabilities
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Given a discrete random variable X, the probability distribution function (pdf)
assigns a probability to each value of X.
The command binompdf(n, p, X) calculates the binomial probability of value X.
Given a random variable X, the cumulative distribution function (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.
The command binomcdf(n, p, X) calculates the binomial probability that the
variable takes on the values of 0 up to (and including) X.
Example – Flipping a Coin
o A fair coin is flipped 6 times.
o Determine the probability that the coin comes up tails exactly 5 times.
o Find the probability that the coin comes up tails at least 1 time.
o Find the probability that the coin comes up tails at most 3 times.
o Construct a pdf table and a pdf histogram for the variable X.
o Construct a cdf table and a cdf histogram for the variable X.
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AP Statistics – Ch 8 – The Binomial and Geometric Distributions
Binomial Formulas
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For any positive whole number n, its factorial n! is
n! = n × (n – 1) × (n – 2) × … × 3 × 2 × 1 and 0! = 1
The number of ways of arranging k successes among n observations is given by
the binomial coefficient
⎛ n⎞
n!
⎜ ⎟=
⎝ k ⎠ k!(n − k)!
Say “binomial coefficient n choose k.” Calculator: MATH Æ PRB Æ nCr
If X is B(n, p), the possible values of X are 0, 1, 2, …, n. If k is any one of these
⎛ n⎞
values, P(X = k) = ⎜ ⎟ p k (1− p) n−k
⎝ k⎠
Example – Flipping a Coin
o A fair coin is flipped 6 times.
o Determine the probability that the coin comes up tails exactly 5 times.
o Find the probability that the coin comes up tails at least 1 time.
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AP Statistics – Ch 8 – The Binomial and Geometric Distributions
Binomial Mean and Standard Deviation
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If X is B(n, p), then the mean is μ = np and the standard deviation is
σ = np(1− p)
As the number of trials n gets larger, the binomial distribution gets close to a
normal distribution. When n is large, we can use normal probability calculations
to approximate hard-to-calculate binomial probabilities.
As a rule of thumb, use the normal approximation when n and p and q satisfy
np ≥ 10 and nq ≥ 10. (note: q is equal to 1 – p)
The accuracy of the normal approximation improves as the sample size n
increases. It is most accurate for any fixed n when p is close to ½ and least
accurate when p is near 0 or 1.
Example – IRS
o The Internal Revenue Service estimates that 8% of all taxpayers filling out
long forms make mistakes. Suppose that a random sample of 10,000
forms is selected. What is the approximate probability that more than 800
forms have mistakes?
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AP Statistics – Ch 8 – The Binomial and Geometric Distributions
Ch 8.2 – The Geometric Distributions
The Geometric Setting
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A situation where these four conditions are satisfied is called a geometric setting.
1. Each observation falls into one of just two categories, which we call “success”
or “failure”.
2. The probability of success, call it p, is the same for each observation.
3. The observations are all independent.
4. The variable of interest is the number of trials required to obtain the first
success.
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If X has a geometric distribution with probability p of success and (1-p) of failure
on each observation, the possible values of X are 1, 2, 3, ….
If n is any one of these values, the probability that the first success occurs on the
nth trial is P(X = n) = (1− p) n−1 p
The probability that it takes more than n trials to see the first success is
P(X > n) = (1− p) n
The mean, or expected value, of X is μ = 1/ p .
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Example – Overweight Americans
o A survey conducted by the Harris polling organization discovered that
80% of all Americans are overweight. Suppose that a number of
randomly selected Americans are weighed.
o How many Americans would you expect to weigh before you encounter
the first overweight individual?
o Find the probability that the fourth person weighed is the first person to be
overweight.
o Find the probability that it takes more than 4 people to observe the first
overweight person.
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