Transcript AP Statistics: Section 8.2 Geometric Probability

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”?
a ) B(5,.1)
1 - P(none)  1 - binomialpdf(5,.1,0)  1 - .59049  .40951
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?
No fixed number of trials
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 Setting
1. Each observation falls into one of just two
categories: _________
failure
success or _________.
2. The n observations are all _______________.
independent
3. The probability of success, call it __,
p is
constant for each observation.
__________
4. *The variable of interest is the number of
trials required to obtain __________________.
the first success
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
1
roll) =
6
P(X = 2) = P(not 3 on
1st
roll and 3 on
2nd
roll) =
5 1 5
 
6 6 36
P(X = 3) = P(not 3 on 1st or 2nd roll and 3 on 3rd roll) =
5 5 1 25
  
6 6 6 216
P(X = 4) = P(not 3 on 1st, 2nd and 3rd roll and 3 on 4th roll) =
5 5 5 1 125
   
6 6 6 6 1296
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:
P(X  n)  (1 - p)
n -1
p
TI83/84:
2nd Vars D : geometpdf ENTER
geometpdf(p, x)
Example: What is the probability that
the 6th girl Spence asks to the prom
will say “yes?”
geometpdf(.1,7)  .053
Construct a probability distribution table for
X = number of rolls of a die until a 3 occurs.
X:
P(X):
1
2
3
4
5
6
7
...
.1667 .1389 .1157 .0965 .0804 .0670 .0558
Note that the number of table entries for X will be
infinite. The probabilities are the terms of a
2
3
geometric sequence, _______________,
a, ar , ar , ar ,..... hence
the name for this random variable.
As with all probability distributions, the sum of
the probabilities must be ___.
1
Recall from Algebra II, maybe Pre-Calculus, that
a
the sum of a geometric sequence is _________.
1 r
So…
p
p
 1
1  (1  p ) p
In the probability histogram, the
first bar represents the probability
of ________.
success 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.
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
1
x 
p
1 p
  2
p
2
x
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?
1
E( X )  x 
 12
1
12
11
1 1
12 
12  132  11.489

2
1
1
144
12
 
P(X > n)
The probability that it takes more than n trials to
n
(
1

p)
see the first success is ________
Example 3: What is the probability that it takes
more than 12 tosses to win a stuffed animal?
12
1
P(X  12)  (1 )  .3520
12