# Random Probability

**Probability**

Work the following probability problems.

Warning: Fractions must be completely reduced in order to receive credit.

- An identification tag consists of three letters followed by six digits. How many different tags can be made if repetitions are allowed?
- How many different arrangements of the letters in the word Honolulu can be made?
- How many
__different__ways can six__different__automobiles be selected from twelve__different__automobiles? - How many ways can a person select 6 candy bars from a list of 10 and 6 salty snacks from a list of 12 to put in a vending machine?
- A restaurant offers 2 choices of meat, 3 choices of potatoes, 6 choices of vegetables, and 3 choices of dessert. How many different possible meals can be made if a customer must select one item from each category?
- The state narcotics bureau must form a 5-member investigative team. If it has 20 agents from which to chose, how many different possible teams can be formed?
- When 2 cards are drawn from a deck, with no replacement, find the probability of getting the following:
- a heart and a king b. two 7’s c. a black card and a jack
- The U. S. Department of Health and Human Services report that 12% of Americans have chronic health problems. If 6 people are selected at random, find the probability that all 6 have chronic health problems.
- How many different ways can 7 computer operators be seated in a row?
- A newspaper advertises 6 different movies, 4 plays, and 5 different sports events for the weekend. If a couple selects 3 activities, find the probability that they select 1 of each.
- In a train yard there are 4 tank cars, 12 boxcars, and 7 flatcars. If a train is to be made up of 10 cars, (not counting the engine) what is the probability that the train is made up of 2 tank cars, 5 boxcars, and 3 flatcars?
- At a convention there are 7 mathematics instructors, 5 computer science instructors, 3 statistics instructors, and 4 science instructors. If an instructor is selected at random, find the probability of getting a science instructor.
- The probability that a given tourist goes to the amusement park is 0.47, and the probability that she goes to the water park is 0.58. If the probability that she goes to either the water park or the amusement park is 0.95, what is the probability that she visits both of the parks on vacation?
- Approximately 12% of the civilian population are veterans. If 5 civilians are chosen at random, what is the probability that at least 1 is a veteran?
- If one card is drawn at random from a standard deck of cards, what is the probability of selecting a king or a black card?
- A random sample of adults was asked what they preferred to watch on TV, sports, comedy, or drama.

** Sports Comedy Drama **

Male 30 15 10

Female 15 10 20

If one adult is selected at random, find these probabilities.

- is a male
- prefers watching sports given that she is a female
- is a male given that he prefers watching sports
- is a female and prefers watching drama
- is a female or prefers watching comedy
- is a male and prefers watching sports
- prefers watching drama given that she is a female
- is a male or prefers watching drama

**Solution**** **

1.

The tag would look like this LLLDDDDDD where L represent letter and D represent digits. Since repetition is allowed, for every L there are 26 options to choose from and for every D there are 10 options to choose from. So, the number of different tags will be:

26 *26*26*10*10*10*10*10*10

= 26 ^{3} * 10 ^{6}

2.

Honolulu

Total letter = 8

O, l and u comes twice.

So, the number of different arrangement will be 8! / (2! * 2! *2!)

= 3*5*6*7*8

= 5040

3.

^{12}C_{6}

4.

^{10}C_{6 *}^{12}C_{6}

5.

Total number of possible meals will be 2*3*6*3 =108

6.

^{20}C_{5}

7.

a.

(13/52) + (4/52) – (16/52) = (1/52)

b.

(4/52) * (3/51) = (1/ 13)*(1/17)

= (1/221)

c.

(26/52) + (4/52) – (28/52) = (2/52)

= (1/26)

8.

Probability that an American have chronic health problems = 0.12. By using binomial theorem the required probability will be

^{6}C_{6} * (0.12)^{6} = 1* (0.12) ^{6}

9.

Total different ways will be 7!

10.

The required probability will be

( ^{6}C_{1} * ^{4}C_{1} * ^{5}C_{1} ) / ( ^{15}C_{3} ) = (12/91)

11.

( ^{4}C_{2} * ^{12}C_{5} * ^{7}C_{3} ) / ( ^{23}C_{10}) = ( 6*8*9*11*5) / (17*19*22*23) = (1080/7429)

12.

The probability of getting a science instructor is (4/19) .

13.

The probability that she visits both of the parks on vacation will be equal to

0.47 +0.58 – 0.95

= 1.05 – 0.95

= 0.10

14.

1 – P(X=0)

=1- [ ^{5}C_{0} * (0.12)^{0} * (0.88) ^{5} ]

= 1-0.53

=0.47

15.

The probability of selecting a king or a black card will be (4/52) + (13/52) – (1/52)

= (16/52) = (4/13)

16.

Total male = 55

Total female = 45

Total person in sports = 45

Total person in Comedy = 25

Total person in Drama = 30

Total person = 55+45 = 100

a.

(55/100) = (11/20)

b.

(15/45) = (1/3)

c.

(30/45) = (2/3)

d.

(20/100) = (1/5)

e.

(60/100) = (3/5)

f.

(30/100) = (3/10)

g.

(20/45) = (4/9)

h.

(75/100) = (3/4)