# Exercises - Counting and Probability Review

1. A quiz has 3 true/false questions and 4 multiple questions (with 4 possible answers each). If one randomly guesses the answer to each question on the quiz, what is the probability of getting all the correct answers?

$\displaystyle{\frac{1}{2048}}$
2. How many ways can you arrange 5 people in a row for a photograph?

$5! = 120$
3. King Art is sitting at his round table (at the head, of course). His seven knights are sitting around him. How many ways can his knights be arranged?

4. How many different arrangements are there of the letters in the following words:

1. LETTERED
2. LEVELED
3. COVINGTON
1. $\displaystyle{\frac{8!}{3! 2!} = 3360}$

2. $\displaystyle{\frac{7!}{2! 3!} = 420}$

5. A jar has 5 red, 6 blue, and 7 green marbles. Some of the marbles are large and the rest are small. Two of the red marbles are large, as are two of the blue marbles, and three of the green marbles. A marble is selected at random. Find the probability that this marble is ...

1. small
2. blue or large
3. red, given that it is small
$$\begin{array}{c|c|c|c|c} & \textrm{red} & \textrm{blue} & \textrm{green} & \textrm{total}\\\hline \textrm{large} & 2 & 2 & 3 & 7\\\hline \textrm{small} & 3 & 4 & 4 & 11\\\hline \textrm{total} & 5 & 6 & 7 & 18 \end{array}$$
1. $\displaystyle{\frac{11}{18}}$

2. $\displaystyle{\frac{11}{18}}$

3. $\displaystyle{\frac{3}{11}}$

6. A jar contains 8 red balls and 4 blue balls. Four balls are drawn at random without replacement.

1. Find the probability that exactly one of the balls is blue.
2. Find the probability that at most one of the balls is blue.
1. $\displaystyle{\frac{({}_4 C_1)({}_8 C_3)}{{}_{12} C_4} = 0.4525}$

2. We need the probability of $0$ or $1$ blue balls, which is given by

$$\displaystyle{\frac{({}_4 C_1)({}_8 C_3)}{{}_{12} C_4} + \frac{({}_4 C_0)({}_8 C_4)}{{}_{12} C_4} = 0.5939}$$
7. A jar has 5 red, 6 blue, and 7 green marbles. Five marbles are selected at random without replacement. Find the probability that there are

1. exactly 3 green marbles
2. at least 3 green marbles

1. $\displaystyle{P(3\textrm{ green}) = \frac{({}_7 C_3)({}_{11} C_2)}{{}_{18} C_5} = 0.225}$

2. $\displaystyle{P(3\textrm{ or }4\textrm{ or }5\textrm{ green}) = \frac{({}_7 C_3)({}_{11} C_2) + ({}_7 C_4)({}_{11} C_1) + {}_7 C_5}{{}_{18} C_5} = 0.272}$

8. A club has 20 members, of which 12 are freshmen, 14 are female, and 8 are female freshmen. A club member is selected at random. Find the probability that the student is

1. a sophomore
2. a freshman or female
3. a freshman, given that she is female
$$\begin{array}{c|c|c|c} & \textrm{Freshmen} & \textrm{Sophomore} & \textrm{Total}\\\hline \textrm{Female} & 8 & 6 & 14\\\hline \textrm{Male} & 4 & 2 & 6\\\hline \textrm{Total} & 12 & 8 & 20 \end{array}$$
1. $\displaystyle{\frac{8}{20}}$

2. $\displaystyle{\frac{18}{20}}$

3. $\displaystyle{\frac{8}{14}}$

9. A fair coin is tossed three times. Write out the sample space and find the probability that the last toss landed heads, given that at least one coin landed heads.

Sample space: $\{HHH,HHT,HTH,THH,TTH,THT,HTT,TTT\}$; probability = $4/7$
10. Two 4-sided dice (labeled 1,2,3,4) are rolled. Write out the sample space for the sum showing on the dice, and find the probability of rolling a sum of 3 or 4.

Sample space: $$\begin{array}{c|c|c|c|c|} & 1 & 2 & 3 & 4\\\hline 1 & 2 & 3 & 4 & 5\\\hline 2 & 3 & 4 & 5 & 6\\\hline 3 & 4 & 5 & 6 & 7\\\hline 4 & 5 & 6 & 7 & 8\\\hline \end{array}$$ $\displaystyle{P(3 \textrm{ or } 4) = \frac{5}{16}}$
11. A club consists of 25 members (freshmen and sophomores) of which 12 are freshmen, 13 are female, and 5 are sophomore males. Find the probability that a randomly selected student is:

1. a sophomore
2. a female freshman
3. a male, given they are a freshman
12. An urn contains 6 red balls and 3 blue balls. Three balls are drawn.

1. If the balls are drawn with replacement, what is the probability of getting all red balls?
2. If the balls are drawn without replacement, what is the probability of getting all red balls?
3. If the balls are drawn without replacement, what is the probability of getting at least two red balls?