Q: A fair die is rolled twice. What is the probability that both rolls result in a 6?

$$ \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} $$

Q: A password consists of 3 letters followed by 2 digits. If letters and digits can be repeated, how many different passwords are possible?

$$ 26^3$ \times 10^2$ = 175,760 $$

Q: A box contains 5 red, 4 blue, and 3 green balls. If 2 balls are drawn at random without replacement, what is the probability that both are red?

$$ \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33} $$

Question 1

Which of the following best describes the Law of Large Numbers?

Accepted Answer

As the number of trials in a random process increases, the proportion of times an event occurs approaches the theoretical probability of the event.

Question 2

A bag contains 3 red, 4 blue, and 5 green marbles. If one marble is drawn at random, what is the probability it is blue? Use the classical method.

Accepted Answer

Question 3

Suppose you flip a fair coin 1000 times. Which of the following is most likely to be true according to the Law of Large Numbers?

Accepted Answer

The proportion of heads will be close to 0.5.

Question 4

A probability model for the number of rooms in a housing unit is given. The probability of a unit having 3 rooms is 0.112. What is the probability that a randomly selected unit has either 3 or 4 rooms, if the probability for 4 rooms is 0.175?

Accepted Answer

$$ 0.112 + 0.175 = 0.287 $$

Question 5

Which of the following is an example of a subjective probability?

Accepted Answer

A doctor estimates there is a 20% chance of recovery based on experience.

Question 6

If $$ P(A) = 0.4 $$ and $$ P(B) = 0.3 $$ and A and B are mutually exclusive, what is $$ P(A 	ext{ or } B) $$?

Accepted Answer

$$ 0.7 $$

Question 7

A survey of 200 households found that 120 own a dog, 80 own a cat, and 30 own both. What is the probability that a randomly selected household owns either a dog or a cat?

Accepted Answer

$$ \frac{120 + 80 - 30}{200} = \frac{170}{200} = 0.85 $$

Question 8

If the probability that a randomly selected person is left-handed is 0.1, what is the probability that a randomly selected person is not left-handed?

Accepted Answer

$$ 1 - 0.1 = 0.9 $$

Question 9

Which of the following statements about independent events is correct?

Accepted Answer

Two events are independent if the occurrence of one does not affect the probability of the other.

Question 10

A fair die is rolled twice. What is the probability that both rolls result in a 6?

Accepted Answer

$$ \frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36} $$

Question 11

A professor has 5 test questions and wants to arrange them in order. How many different ways can the questions be arranged?

Accepted Answer

$$ 5! = 120 $$

Question 12

How many ways can a committee of 3 be chosen from 8 people?

Accepted Answer

$$ \binom{8}{3} = 56 $$

Question 13

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

Accepted Answer

$$ 26^3$ 	imes 10^2$ = 175,760 $$

Question 14

A box contains 5 red, 4 blue, and 3 green balls. If 2 balls are drawn at random without replacement, what is the probability that both are red?

Accepted Answer

$$ \frac{5}{12} 	imes \frac{4}{11} = \frac{20}{132} = \frac{5}{33} $$

Question 15

A survey found that 60% of people have high-speed internet and 30% have land-line phones. If 20% have both, what is the probability that a randomly selected person has either high-speed internet or a land-line phone?

Accepted Answer

$$ 0.6 + 0.3 - 0.2 = 0.7 $$

Question 16

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

Accepted Answer

$$ 0.4 	imes 0.5 = 0.2 $$

Question 17

A family has three children. What is the probability that all are girls, assuming each child is equally likely to be a boy or a girl?

Accepted Answer

$$ (\frac{1}{2})^3$ = \frac{1}{8} $$

Question 18

A standard deck of 52 cards is shuffled. What is the probability that the first card drawn is a heart or a king?

Accepted Answer

$$ \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} $$

Question 19

How many different ways can the letters in the word 'STATISTICS' be arranged?

Accepted Answer

$$ \frac{10!}{3!3!2!1!1!} = 50400 $$

Question 20

A simulation is used to estimate the probability of an event. Which of the following statements is true?

Accepted Answer

The estimated probability will get closer to the true probability as the number of simulations increases.

Chapter 5: Probability – Rules, Methods, and Applications

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