Page 148 - Probability Demystified

CHAPTER 8 Other Probability Distributions 137
3 1
2
1 1 1 1 1
Using the formula given above, 1 ¼ ¼
2 2 2 2 8

EXAMPLE: A die is rolled. Find the probability of getting the first three on
the fourth roll.

SOLUTION:
1 4 1 1 5 3 1 125
1
Let p ¼ and n ¼ 4; hence, 1 ¼ ¼ 0:096
6 6 6 6 6 1296
The geometric distribution can be used to answer the question, ‘‘How long
on average will I have to wait for a success?’’
Suppose a person rolls a die until a five is obtained. The five could occur
on the first roll (if one is lucky), on the second roll, on the third roll, etc. Now
the question is, ‘‘On average, how many rolls would it take to get the first
five?’’ The answer is that if the probability of a success is p, then the average
1
or expected number of independent trials it would take to get a success is .
p
1
In the dice situation, it would take on average 1 ‚ or 6 trials to get a five.
6
This is not so hard to believe since a five would occur on average one time in
1
every six rolls because the probability of getting a five is .
6
EXAMPLE: A coin is tossed until a head is obtained. On average, how many
trials would it take?

SOLUTION:
1
Since the probability of getting a head is , it would take 1 ‚ p trials.
2
1 2
1 ¼ 1 ¼ 2
2 1
On average it would take two trials.

Now suppose we ask, ‘‘On average, how many trials would it take to get
two fives?’’ In this case, one five would occur on average once in the next six
trials, so the second five would occur on average once in the next six trials.
In general we would expect k successes on average in k/p trials.

EXAMPLE: If cards are selected from a deck and replaced, how many trials
would it take on average to get two clubs?