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Math 166, Fall 2015, Robert
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2.3- Probability Applications of Counting
Distinguishable Permutations: If a set of n objects consists of k groups
of size n1 , n2 , . . . , nk with all of the objects in each group being identical and
n1 + n2 + · · · + nk = n, then the number of distinguishable permutations of these
objects is
n!
n1 !n2 ! · · · nk !
Example 1 A grocery store wants to tempt customers into buying more produce
by displaying a single row of ripe fruit near the entrance. The manager of the
store sets aside 4 apples, 6 vines of grapes, 3 bunches of bananas, and 5 peaches
to make the display and tells the employees that they must use all of the set
aside fruit. How many different arrangements can the employees make?
Example 2 How many different ways can we rearrange the letters of the word
“bookkeeper”?
We can use all of the counting techniques we have developed so far to calculate
the probability of events happening. Recall that if S is a uniform sample space
and E is an event in S, then
P (E) =
1
n(E)
n(S)
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Math 166, Fall 2015, Robert
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Example 3 A fair die is rolled six times. What is the probability that every
one of the six numbers will be obtained?
Example 4 In the card game poker, a player is dealt a five card hand from a
deck of 52 cards. Some hands are more valuable than others, and the relative
value of these hands depends on the probability of getting them. Assuming that
any five card hand is equally likely, find the probability of each of the following
hands.
• Straight Flush (5 cards in a sequence, all the same suit)
• Flush (all cards are the same suit) that is not a straight
• Straight (5 cards in a sequence) that is not a flush
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Math 166, Fall 2015, Robert
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Example 5 A company produces safety glass that is designed to break in a
fashion that minimizes the chance of injury when someone hits it. In a shipment
of 30 pieces of this glass, 5 are randomly selected to be tested by quality control.
If the shipment has 8 defective pieces of glass, what is the probability that quality
control will discover that defective pieces are present?
Example 6 A number of people are gathered together in the same room. Assuming that there are 365 days in a year and that each person is equally likely
to have their birthday on any given day, what is the probability that at least two
people share the same birthday if
• There are 5 people
• There are 23 people
• Including the instructor, 62 people in this class had their birthday recorded.
What is the probability that at least two of them share a birthday?
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