Probability
Chapter 11
1
Counting Techniques
and
Probability
Section 11.2
2
Examples
Helen and Patty both belong to a club of 25
members. A committee of 4 is to be selected
at random from the 25 members. Find the
probability that both Helen and Patty will be
selected.
2nCr2  23nCr2
P ( both selected) 
25nCr4
253

12,650
1

50
3
Examples
A piggy bank contains 2 quarters, 3 nickels, and 2
dimes. A person takes 2 coins at random from this
bank. Label the coins Q1, Q2, N1, N2, N3, D1, and D2
so they can all be regarded as different. Then find
the probabilities that the values of the 2 coins
selected are the following:
a. 35¢
b. 50¢
2nCr1  2nCr1 2  2
4
a. P (35¢ ) 


7 nCr 2
21
21
2nCr 2
1
b. P (50¢ ) 

7 nCr 2 21
4
Examples
Assume that 2 cards are drawn in succession and
without replacement from an ordinary deck of 52
cards. Find the probability that
a. 2 kings are drawn.
b. 1 spade and 1 king other than the king of
spades (in that order) are drawn.
4nPr 2
43
12
1
a. P(2 kings) 



52nPr 2 52  51 2652 221
13 3
39
1
b. P(1spade1king) 



52 51 2652 68
5
Examples
If 2% of the auto tires manufactured by a
company are defective and 2 tires are randomly
selected from an entire week’s production, find
the probability that neither is defective.
P(neither defective )  (.98)(.98)  (.98)2  .9604
Find the probability of at least 1 of the 2
selected tires is defective.
P(at least 1 defective )  1  P(none defective )
 1  .9604
 0.0396
6
Example
A box contains 10 computer disks and 2 are
defective. If 3 disks are randomly selected
From the box, find the probability that exactly
2 are defective.
defective
good
2nCr2  8nCr1
8
1
P(2 defective ) 


10nCr3
120 15
total
7
Examples
A hat contains 24 names, 13 of which are female.
If seven names are randomly drawn from the hat,
what is the probability that at least one male name
is drawn?
13n Pr 7
1
 0.9950
24n Pr 7
In a sample of 32 hand-held calculators, 26 are
known to be nonfunctional. If 21 of these calculators
are selected at random, what is the probability that
exactly 17 in the selection are nonfunctional? Round
to the nearest thousandth.
6nCr4  26nCr17
 0.363
32nCr21
8
Example
When printing color inserts for newspapers, it
sometimes happens that the registration of the print
colors is imperfect. (This results in the different colors
not being aligned properly, so the image is blurry.)
Suppose that in a run of 1217 one-page inserts, 78
have registration errors. If 4 inserts are chosen at
random, what is the probability that at least one of
them has a color registration error?
(1217  78)n Pr 4
1
 0.2330
1217 n Pr 4
9
Example
 A bag contains 5 red marbles, 6 blue
marbles, 11 white marbles and 9 yellow
marbles. You are asked to draw 5 marbles
from the bag without replacement. What is
the probability of drawing less than two
yellow marbles?
22nCr5  9nCr0  22nCr4  9nCr1
P(less than 2 yellow) 
31nCr5
26,334 65,835
169,911
92,169 1463


169,911 2697

END
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