# Probability, Conditional Probability, Permutations, Solution KEY

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```					Probability, Conditional Probability, Permutations, Solution KEY REVISED!!
Combinations, Pascal’s Triangle REVIEW (1.6, 9.7, 6.7, 6.8, 12.2-12.4)

1. Given the following data, find the mean, median, mode, range and standard deviation.
10 10 9 8 8 8 7 7 7 7 7 7 7 6 6 6 5 5 4 2 1
mean: 6.5, mode: 7, median: 7, range: 9, standard deviation: 2.3
2. In how many ways can a motel chain select three sites for the construction of new motels if 14 sites are
available?
14C3 = 364
3. Find the number of 8-letter permutations of all the letters in the word NOVEMBER that end with ER.
6! = 720
4. Two Cards are drawn from a standard 52-card deck without replacement.
What is the probability of drawing:
a) a six and then a face card? (4/52)(12/51) = 4/221
b) two red cards?      (26/52)(25/51) = 25/102
c) a heart, given that the first cards is not a heart? 13/51
d) a heart, given that the first cards is a heart? 12/51
e) an ace and then a spade?         (1/52)(12/51) + (3/52)(13/51) = 1/52

5. Solve for n:   n   C 2  55 (Only an algebraic solution will be accepted.)
n!
By definition:                  55
(n  2)!2!

1  2  3  ..... (n  2)(n  1)n
Expand:                                           55
1  2  3  ..... (n  2)  1  2

(n  1)n
Simplify:             55
1 2

Cross Multiply: n 2  n  2  55 …. Now set = 0 and solve…… n 2  n  110  0 , (n  11)(n  10)  0
So n = 11 or -10 and n cannot be negative, so n = 11
6. Use Pascal’s Triangle to expand (ax 2  3 y 3 ) 4 .
Answer: a 4 x8  12a 3 x 6 y 3  54a 2 x 4 y 6  108ax 2 y 9  81y12
7. How many license plates starting with three letters followed by three numbers if the last number must be
odd and the “Q” cannot be used.
(25)(25)(25)(10)(10)(10)(5) = 78,125,000
(n  2)!
8. Find                Answer: n(n  1)(n  2)  n(n 2  3n  2)  n3  3n 2  2n
(n  1)!
9. When Kimberly bought her new car, she found that there were 72 different ways her car could be equipped.
Her choices included four choices of engine and three choices of transmission. If her only other choice was
color, how many choices of color did she have?
72 = (4)(3)x, 72 = 12x, x = 6
10. In how many ways can Dorothy invite two girls and three boys to a party if she chooses from eight girls and
six boys?      (8C2)(6C3) = 560
11. Cynthia arrives at a party very late. The only things left on the buffet table are 6 turkey rollups, 4 mini
quiches, and 5 crab cakes. Cynthia eats 3 items. What is the probability that she ate 2 turkey rollups and 1
6 5 5         5
crab cake? Answer:            
15 14 13 91
12. Given the following chart showing the majors of students at a small technical college, find each of the
following
probabilities:                            Freshmen    Sophomores        Juniors         Seniors        TOTALS
Architecture          50           30              40              25            145
Business              60           55              45              30            190
Engineering           40           35              50              55            180
150          120             135             110            515
a. P(Sophomore)                       b. P(Engineering major | Freshman)
120 24                                 40     4
Answer:                                 Answer:      
515 103                               150 15
c. P(Freshmen | Business Major)               d. P(Business or Architecture Major | Senior)
60     6                              55 1
Answer:                                 Answer:      
190 19                                 110 2
13. A 10-item multiple choice test is given. Each problem has 4 choices. If Sally gets #1, 2 and 10 correct, how
many ways can she answer the test?
(1) (1) (4) (4) (4) (4) (4) (4) (4) (1) =
14. Two cards are drawn at random from a standard deck of 52 playing cards. The first card is NOT returned to
the deck before the second card is drawn. Find the probability that the first card is a face card and the
second card is red. (6/52)( 25/51) + (6/52)( 26/51) = 3/26
15. What is the probability of selecting an ACE followed by a JACK from a standard deck of 52 cards if the
first card IS replaced before the second card is drawn?
(4/52)( 4/52) = 1/169

16. The local Family Restaurant has a daily breakfast special in which the customer may choose one item from
each of the following groups:
Breakfast Sandwich              Accompaniments                    Juice
Egg and Ham                      Potatoes                    Orange
Egg and Bacon                   Apple slices                 Cranberry
Egg and Cheese                    Fresh Fruit                  Tomato
Pastry                      Apple
Grape
a) How many different breakfast specials are possible?               (3) (4) (5) = 60
b) How many different breakfast special without meat are possible? (1) (4) (5) = 20
17. From a group of 16 Juniors and 14 Sophomores, how many different committees of 8 can be formed if at
least 5 are Juniors? 16C5 . 14C3 + 16C6 . 14C2 + 16C7 . 14C1 + 16C8 = 2,491,710

18. How many permutations of the letters of the word BABBLING are there? 8!/3! = 6,720

19. Find the number of 5-card hands that can be obtained from a standard deck of 52 cards if the even cards
(2,4,6,8,10’s) are removed. 32C5 = 201,376

20. In how many ways can 6 teachers and 4 students stand in a line if the students must stand together?

7(4!)(6!) = 120,960

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