Q: Given the probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3$$, what is the mean $$\mu of $$the distribution?

$$0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Q: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

$$\binom{5}{3} (0.5)^3$ (0.5)^2$ = 10 \times 0.125 \times 0.25 = 0.3125$$

Q: If a binomial distribution has $$n = 20$$ and $$p = 0.3$$, what is the variance $$\$$sigma^2$$?

$$20 \times 0.3 \times 0.7 = 4.2$$

Q: A box contains 8 light bulbs, 3 of which are defective. If 2 bulbs are selected at random without replacement, what is the probability that both are defective?

$$\dfrac{3}{8} \times \dfrac{2}{7} = \dfrac{6}{56} = \dfrac{3}{28}$$

Q: A discrete probability distribution has $$\mu = 2$$ and $$P(0) = 0.1$$, $$P(1) = 0.2$$, $$P(2) = 0.3$$, $$P(3) = 0.4. $$What is the variance $$\$$sigma^2$$?

$$(0-2)^2$$ \times 0.1 + $$(1-2)^2$$ \times 0.2 + $$(2-2)^2$$ \times 0.3 + $$(3-2)^2$$ \times 0.4 = 0.4 + 0.2 + 0 + 0.4 = 1.0$$

Q: A hospital records the number of births per day as follows: 0 (probability 0.05), 1 (0.15), 2 (0.4), 3 (0.3), 4 (0.1). What is the standard deviation $$\sigma of $$the distribution? (Mean $$\mu = 2.25)$$

$$\sqrt{(0-2.25)^2$ \times 0.05 + (1-2.25)^2$ \times 0.15 + (2-2.25)^2$ \times 0.4 + (3-2.25)^2$ \times 0.3 + (4-2.25)^2$ \times 0.1} \approx 0.97$$

Question 1

A password consists of 3 letters followed by 2 digits. If letters and digits can be repeated, how many different passwords are possible? (Assume 26 letters and 10 digits.)

Accepted Answer

$$26^3$$ 	imes $$10^2$$

Question 2

A bag contains 5 red balls and 7 blue balls. If one ball is drawn at random, what is the probability that it is blue?

Accepted Answer

$$\dfrac{7}{12}$$

Question 3

Given the following frequency distribution for the number of pets per household: 0 pets (10 households), 1 pet (20 households), 2 pets (15 households), 3 pets (5 households). What is the probability that a randomly selected household has at least 2 pets?

Accepted Answer

$$\dfrac{15+5}{50} = \dfrac{20}{50} = 0.4$$

Question 4

A survey shows that 60% of students own a laptop and 30% own a tablet. If 20% own both, what is the probability that a randomly selected student owns either a laptop or a tablet?

Accepted Answer

$$0.6 + 0.3 - 0.2 = 0.7$$

Question 5

A card is drawn from a standard deck of 52 cards. What is the probability that it is a heart, given that it is red?

Accepted Answer

$$\dfrac{13}{26} = 0.5$$

Question 6

If $$P(A) = 0.4$$ and $$P(B|A) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$0.4 	imes 0.5 = 0.2$$

Question 7

How many ways can a committee of 4 be chosen from 10 people?

Accepted Answer

$$\dfrac{10!}{4!6!}$$

Question 8

Which of the following is a discrete random variable?

Accepted Answer

The number of cars sold by a dealership in a day

Question 9

Given the probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3$$, what is the mean $$\mu of $$the distribution?

Accepted Answer

$$0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 10

A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Accepted Answer

$$\binom{5}{3} (0.5)^3$ (0.5)^2$ = 10 	imes 0.125 	imes 0.25 = 0.3125$$

Question 11

If a binomial distribution has $$n = 20$$ and $$p = 0.3$$, what is the variance $$\$$sigma^2$$?

Accepted Answer

$$20 	imes 0.3 	imes 0.7 = 4.2$$

Question 12

A restaurant offers 3 appetizers, 5 main courses, and 4 desserts. How many different meals (one appetizer, one main course, one dessert) can be ordered?

Accepted Answer

$$3 	imes 5 	imes 4 = 60$$

Question 13

A box contains 8 light bulbs, 3 of which are defective. If 2 bulbs are selected at random without replacement, what is the probability that both are defective?

Accepted Answer

$$\dfrac{3}{8} 	imes \dfrac{2}{7} = \dfrac{6}{56} = \dfrac{3}{28}$$

Question 14

Which formula would you use to find the probability of getting at least one success in $$n$$ independent Bernoulli trials?

Accepted Answer

$$1 - (1-p)^n$$

Question 15

A lottery requires you to pick 6 numbers from 1 to 49. How many different combinations of 6 numbers are possible?

Accepted Answer

$$\dfrac{49!}{6!43!}$$

Question 16

Which of the following is a continuous random variable?

Accepted Answer

The time it takes for a chemical reaction to complete

Question 17

If $$P(A) = 0.7$$, what is the probability that event $$A$$ does not occur?

Accepted Answer

$$1 - 0.7 = 0.3$$

Question 18

A discrete probability distribution has $$\mu = 2$$ and $$P(0) = 0.1$$, $$P(1) = 0.2$$, $$P(2) = 0.3$$, $$P(3) = 0.4. $$What is the variance $$\$$sigma^2$$?

Accepted Answer

$$(0-2)^2$$ 	imes 0.1 + $$(1-2)^2$$ 	imes 0.2 + $$(2-2)^2$$ 	imes 0.3 + $$(3-2)^2$$ 	imes 0.4 = 0.4 + 0.2 + 0 + 0.4 = 1.0$$

Question 19

A hospital records the number of births per day as follows: 0 (probability 0.05), 1 (0.15), 2 (0.4), 3 (0.3), 4 (0.1). What is the standard deviation $$\sigma of $$the distribution? (Mean $$\mu = 2.25)$$

Accepted Answer

$$\sqrt{(0-2.25)^2$ 	imes 0.05 + (1-2.25)^2$ 	imes 0.15 + (2-2.25)^2$ 	imes 0.4 + (3-2.25)^2$ 	imes 0.3 + (4-2.25)^2$ 	imes 0.1} \approx 0.97$$

Probability and Discrete Probability Distributions: Exam 2 Study Guide

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