# Exercises - Probability

1. Toss 3 coins. Find the probability that at least one head shows by writing out the sample space

Sample space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}; probability of at least one head is then $7/8=0.875$

2. Is it unusual to get 3 heads when 3 coins are tossed?

The probability of 3 heads is $1/8 = 0.125$, which is not terribly unusual. (Typically, events that have probability less than $0.05$ are considered unusual.)

3. In rolling a single die, let $A$ represent rolling an even value and $B$ represent rolling a value greater than 4. Find $P(A \textrm{ or } B)$ in two ways: a) using the addition rule and b) considering the sample space.

There are 4 out of 6 values in the sample space that fit this criteria (i.e., $\{2,4,5,6\}$), so the probability is $4/6 = 2/3$.

Using the addition rule, we have: $P(\textrm{even}) + P(\gt4) - P(\textrm{even and } \gt 4) = 3/6 + 2/6 - 1/6 = 4/6 = 2/3$

4. In rolling a single die, let $A$ represent rolling an odd value and $B$ represent rolling a $6$. Find $P(A \textrm{ and } B)$ in two ways: a) using the multiplication rule and b) considering the sample space.

A and B can't both happen. A number can't be both odd and $6$. So the probability is $0$.

5. A single card is drawn from a deck. Find the probability of selecting:

1. a 4 or a diamond
2. a club or a diamond
3. a jack or a black card
1. $\displaystyle{\frac{4}{52} + \frac{13}{52} - \frac{1}{52} \doteq 0.3077}$

2. $\displaystyle{\frac{13}{52} + \frac{13}{52} = 0.5}$

3. $\displaystyle{\frac{4}{52} + \frac{26}{52} - \frac{2}{52} \doteq 0.5385}$

6. At a used-book sale, 100 books are adult books and 160 are children's books. Of the adult books, 70 are nonfiction while 60 of the children's books are nonfiction. If a book is selected at random, find the probability that it is:

1. fiction
2. not a children's nonfiction book
3. an adult book or a children's nonfiction book
1. $\displaystyle{\frac{130}{260} = 0.5}$

2. $\displaystyle{1 - \frac{60}{260} \doteq 0.2308}$

3. $\displaystyle{\frac{100}{260} + \frac{60}{260} \doteq 0.6154}$

7. When two dice are rolled, find the probability of getting a sum that is

1. 5 or 6
2. greater than 9
3. less than 4 or greater than 9
4. divisible by 4
1. $\displaystyle{\frac{4+5}{36} = \frac{1}{4}}$

2. $\displaystyle{\frac{3+2+1}{36} = \frac{1}{6}}$

3. $\displaystyle{\frac{1+2+3+2+1}{36} = \frac{1}{4}}$

4. $\displaystyle{\frac{3+5+1}{36} = \frac{1}{4}}$

8. Suppose two coins are tossed, and you know that at least one of them resulted in "tails". What is the probability that they are both tails?

$\displaystyle{\frac{1}{3}}$

9. In drawing two cards from a standard deck, what is the probability of drawing an ace on the first draw and a king on the second draw?

$\displaystyle{\frac{4}{52} \cdot \frac{4}{51} \doteq 0.0060}$

10. Roll a single die. What is the probability that one rolls a 1 or 2 given that one rolled an even value?

$\displaystyle{\frac{1}{3}}$

11. At a large university, the probability that a student takes calculus and is on the dean's list is 0.042. The probability that a student is on the dean's list is 0.21. Find the probability that the student is taking calculus given that he or she is on the dean's list.

$\displaystyle{\frac{0.042}{0.21} = 0.2}$

12. Eighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results of the survey are shown in the table below: $$\begin{array}{l|c|c|c} & \textrm{Favor} & \textrm{Oppose} & \textrm{No Opinion}\\\hline \textrm{Freshmen} & 15 & 27 & 8\\\hline \textrm{Sophomore} & 23 & 5 & 2\\\hline \end{array}$$ If a student is selected at random, find the probability that

1. given the student is a freshman, he or she opposes the ban
2. given the student favors the ban, he or she is a sophomore
1. $\displaystyle{\frac{27}{15+27+8} = 0.54}$

2. $\displaystyle{\frac{23}{15+23} \doteq 0.6053}$

13. Find the probability that if a coin is tossed twice, the first toss is "heads", while the second is "tails"

$\displaystyle{\frac{1}{2} \cdot \frac{1}{2} = 0.25}$

14. If 37% of high school students said that they exercise regularly, find the probability that 5 randomly selected high school students will say that they exercise regularly.

$\displaystyle{(0.37)^5 \doteq 0.0069}$

15. If 2 cards are selected from a standard deck of 52 cards without replacement, find the probability that

2. both are the same suit
1. $\displaystyle{\frac{13}{52} \cdot \frac{12}{51} \doteq 0.0588}$

2. $\displaystyle{\frac{12}{52} \doteq 0.2308}$

16. The U.S. Department of Health and Human Services reports that 15% of Americans have chronic sinusitis. If 5 people are selected at random, find the probability that at least one has chronic sinusitis.

Use the complement. $\quad 1 - (0.85)(0.85)(0.85)(0.85)(0.85) = 1 - 0.85^5 \doteq 0.5563$

17. A jar contains 8 red marbles, 9 blue marbles, and 10 green marbles. Four marbles are chosen at random without replacement. Find the probability of getting

1. all green marbles
2. 2 red and 2 blue marbles
3. no green marbles
4. exactly 2 green marbles
5. at most 2 green marbles
6. marbles that are all the same color
1. $\displaystyle{\frac{10}{27} \cdot \frac{9}{26} \cdot \frac{8}{25} \quad \textrm{or} \quad \frac{({}_{10} C_4)}{({}_{27} C_4)}}$

2. $\displaystyle{\frac{({}_8 C_2)({}_9 C_2)}{({}_{27} C_4)}}$

3. $\displaystyle{\frac{17}{27} \cdot \frac{16}{27} \cdot \frac{15}{27} \quad \textrm{or} \quad \frac{({}_{17} C_4)}{({}_{27} C_4)}}$

4. $\displaystyle{\frac{({}_{10} C_2)({}_{17} C_2)}{({}_{27} C_4)}}$

5. $\displaystyle{\frac{({}_{10} C_0)({}_{17} C_4)}{({}_{27} C_4)} + \frac{({}_{10} C_1)({}_{17} C_3)}{({}_{27} C_4)} + \frac{({}_{10} C_2)({}_{17} C_2)}{({}_{27} C_4)}}$

6. $\displaystyle{\frac{({}_{8} C_4)}{({}_{27} C_4)} + \frac{({}_{9} C_4)}{({}_{27} C_4)} + \frac{({}_{10} C_4)}{({}_{27} C_4)}}$

18. An automobile manufacturer has three factories: A, B, and C. They produce 50%, 30%, and 20% respectively, of a specific model of car. 30% of the cars produced in factory A are white, 40% of those produced in factory B are white, and 25% produced in factory C are white.

1. If an automobile produced by the company is selected at random, find the probability that it is white.
2. Given that an automobile selected at random is white, find the probability that it came from factory B.

19. Two manufacturers supply blankets to emergency relief organizations. Manufacturer A supplies 3000 blankets and 4% are irregular in workmanship. Manufacturer B supplies 2400 blankets and 7% are found to be irregular. Given that a blanket is irregular, find the probability that it came from manufacturer B.

20. The Birthday Problem. Find the probability that 2 or more people in a group of $n$ people have the same birthday.

21. In the game of Craps, a player rolls a pair of dice. If, on this first roll, the sum of the dice is 7 or 11, they win immediately. If the sum of the dice is 2, 3, or 12, they lose immediately. If instead, they roll some other value $x$ (called the "point"), they continue to roll the dice until either the point is rolled (where the player then wins), or a 7 is rolled (where the player then loses). What is the probability that the player wins a game of Craps?

22. In rolling a pair of dice, what is the probability that you roll

1. at least a sum of 7?
2. at most a 7 given that at least one die shows a 3

23. Toss a coin 3 times. What is the probability that

1. there is at most one head
2. one gets 2 heads given that the first toss is a tail

24. For a bridge hand of 13 cards, find the probability that you have