Math 141: Business Mathematics I
Fall 2015
§7.4 Use of Counting Techniques in Probability
Instructor: Yeong-Chyuan Chung
So far, we have been dealing with experiments with relatively small numbers of possible
outcomes. For such experiments, we may choose to list all the possible outcomes, and count
how many of them are in the event that we are considering, thereby obtaining a ratio that
tells us what the probability of that event is. When the number of possible outcomes is
large, this listing method will not be practical but we can use the counting techniques from
§6.3 and §6.4 to help us to compute probabilities.
In this section, we will restrict our attention to uniform sample spaces. Recall that these are
sample spaces in which the outcomes are equally likely. In this case, the probability of an
event E is given by
n(E)
P (E) =
.
n(S)
Example (Exercise 10, 12 in the text). Four balls are selected at random without replacement
from an urn containing three white balls and five blue balls. Find the probability that:
(a) All of the balls are blue.
(b) Two or three of the balls are white.
Example (Exercise 20 in the text). A customer at Cavallaro’s Fruit Stand picks a sample
of 3 oranges at random from a crate containing 60 oranges, of which 4 are rotten. What is
the probability that the sample contains 1 or more rotten oranges?
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§7.4 Use of Counting Techniques in Probability
Example (Exercise 28 in the text). In the game of blackjack, a 2-card hand consisting of
an ace and a face card or a 10 is called a blackjack. (In a standard deck of cards, there are
4 suits with 13 cards each, namely A, 2, 3, ... , 10, J, Q, K. Among these cards, J, Q, and
K, are face cards.)
(a) If a player is dealt 2 cards from a standard deck of 52 well-shuffled cards, what is the
probability that the player will receive a blackjack?
(b) If a player is dealt 2 cards from 2 well-shuffled standard decks (1 card from each deck),
what is the probability that the player will receive a blackjack?
Example (Exercise 26 in the text). Four different written driving tests are administered
by the Motor Vehicle Department. One of these four tests is selected at random for each
applicant for a driver’s license. If a group consisting of two women and three men apply for
a license, what is the probability that:
(a) Exactly two of the five will take the same test?
(b) The two women will take the same test?
§7.4 Use of Counting Techniques in Probability
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Example (Birthday Problem). If a group of 5 people is selected at random, what is the
probability that at least two of them have the same birthday? (Assume that there are 365
days in a year, and that none of the people was born on Feb 29 of a leap year.)
Example. What if a group of 50 people is selected in the previous example instead of 5
people?