AP Statistics: Section 8.2

Geometric Probability

Spence has trouble getting girls to say “Yes” when he asks them for a date. In fact, only 10% of the girls he asks actually agree to go out with him. Suppose

that p = 0.10 is the probability that any randomly selected girl, assume independence, will agree to out with him. Spence desperately wants a date for the prom. (a) What is the probability that at least one of the first 5 girls

asked will say “yes”. (b) How many girls can he expect to ask before the first one says “yes”?

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B(5,.1) )

a

In (b) we will let X = the number of times Spence needs to ask a girl for a date before a girl accepts. Why is this not a binomial

distribution?

trialsofnumber fixed No

A random variable that counts the number of trials needed to obtain

one success is called geometric and the distribution produced by this random variable is called a

geometric distribution.

The Geometric Setting1. Each observation falls into one of just two categories: _________ or _________.

2. The n observations are all _______________.

3. The probability of success, call it __, is __________ for each observation.

4. *The variable of interest is the number of trials required to obtain __________________.

success failure

tindependen

pconstant

successfirst the

Example 1: Consider rolling a single die.

X = the number of rolls before a 3 occurs.

Is this a geometric setting?

Yes

P(X = 1) = P(3 on 1st roll) =

P(X = 2) = P(not 3 on 1st roll and 3 on 2nd roll) =

P(X = 3) = P(not 3 on 1st or 2nd roll and 3 on 3rd roll) =

P(X = 4) = P(not 3 on 1st, 2nd and 3rd roll and 3 on 4th roll) =

6

1

36

5

6

1

6

5

216

25

6

1

6

5

6

5

1296

125

6

1

6

5

6

5

6

5

Rule for Calculating Geometric Probabilities

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, . . . .

8If n is any one of these values, the probability that the first success occurs on the nth trial is:

pp)-(1n)P(X 1-n

TI83/84:

x)p,geometpdf(

ENTER geometpdf:D Vars 2nd

Example: What is the probability that the 6th girl Spence asks to the prom will say “yes?”

.053.1,7)geometpdf(

Construct a probability distribution table for X = number of rolls of a die until a 3 occurs.

Note that the number of table entries for X will be infinite. The probabilities are the terms of a

geometric sequence, _______________, hence the name for this random variable.

X: 1 2 3 4 5 6 7 . . .

P(X): .0965 .1157 .1389 1667. .0558 .0670 0804.

,.....,,, 32 ararara

As with all probability distributions, the sum of the probabilities must be ___.

Recall from Algebra II, maybe Pre-Calculus, that

the sum of a geometric sequence is _________. So…

1

r

a

1

1)1(1

p

p

p

p

In the probability histogram, the first bar represents the probability

of ________. The height of all subsequent bars is smaller since you are multiplying by a number less than 1. So the histogram will

be _____-skewed. Always.

success

right

The Mean and Standard Deviation of the Geometric Random Variable

If X is a geometric random variable with

probability of success p on each trial, then

22x

1

1

p

p

px

Example 2: A game of chance at the state fair involves tossing a coin into a saucer. You win a stuffed animal if the coin lands in and stays on the saucer. A person wins on average 1 out of every 12 times she/he plays. What is the expected number of

tosses for a win? What is the standard deviation?

12

1211

)( xXE

489.11132

1441

1211

121

1211

2

P(X > n) The probability that it takes more than n trials to

see the first success is ________

Example 3: What is the probability that it takes more than 12 tosses to win a stuffed animal?

np)1(

3520.)121-(112)P(X 12