Page 149 - Probability Demystified

138 CHAPTER 8 Other Probability Distributions

SOLUTION:

1
Since there are 13 clubs in a deck of 52 cards, PðclubÞ¼ 13 ¼ . The expected
52
4
1
number of trials for selecting two clubs would be 2 1 or 2 ¼ 2 4 ¼ 8
4
trials. 4
This type of problem uses what is called the negative binomial distribution,
which is a generalization of the geometric distribution.
Another interesting question one might ask is, ‘‘On average how many
rolls of a die would it take to get all the faces, one through six, on a die?’’ In
this case, the first roll would give one of the necessary numbers, so the
probability of getting a number needed on the first roll would be one. On the
5
second roll, the probability of getting a number needed would be since there
6
are 5 remaining needed numbers. The average number of rolls would be
6
ð 1 Þ or . Since two numbers have been obtained, the probability of getting
5=6 5
4
6
the next number would be . The average number of rolls would be ð 1 Þ or .
6 4=6 4
This would continue until all numbers are obtained. So the average number
of rolls it would take to get all the numbers, one through six, would be
6
6
6
6
6
1 þ þ þ þ þ ¼ 14:7. Hence on average it would take about 14.7 rolls
5 4 3 2 1
to get all the numbers one through six.
EXAMPLE: A children’s cereal manufacturer packages one toy space craft in
each box. If there are 4 different toys, and they are equally distributed, find
the average number of boxes a child would have to purchase to get all four.
SOLUTION:
2
3
1
The probabilities are 1, , , and . The average number of boxes for each
4
4
4
4
4
4
are 1 , 1 , 1 , and 1 so the total is 1 þ þ þ ¼ 8 1 which would
1 ð3=4Þ ð2=4Þ ð1=4Þ 3 2 1 3
mean a child on average would need to purchase 9 boxes of cereal since he
or she could not buy 1 of a box.
3
PRACTICE
1. A card from an ordinary deck of cards is selected and then replaced.
Another card is selected, etc. Find the probability that the first club
will occur on the third draw.
2. A die is tossed until a one or a two is obtained. Find the expected
number of tosses.
3. On average how many rolls of a die will it take to get 3 fours?