Q: Suppose you randomly select a card from a standard deck of 52 cards. Let event A be 'the card is a heart' and event B be 'the card is a king.' What is the probability that the card is a heart or a king?

$$P(A \text{ or } B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52}$$

Q: Which of the following is the correct formula for the probability of the union of two events A and B?

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

Q: A bag contains 6 red balls and 4 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

$$\frac{6}{10} \times \frac{5}{9} = \frac{30}{90} = \frac{1}{3}$$

Q: A factory uses two machines to produce widgets. Machine A produces 60% of the widgets and has a defect rate of 2%. Machine B produces 40% of the widgets and has a defect rate of 5%. What is the probability that a randomly selected widget is defective?

$$0.60 \times 0.02 + 0.40 \times 0.05 = 0.012 + 0.02 = 0.032$$

Q: A new medication is tested on 100 patients. 30 experience side effects. If two patients are selected at random with replacement, what is the probability both experience side effects?

$$\frac{30}{100} \times \frac{30}{100} = 0.09$$

Q: A quality control inspector selects two light bulbs at random from a box of 100 bulbs, 10 of which are defective. What is the probability that both bulbs are defective if the selection is without replacement?

$$\frac{10}{100} \times \frac{9}{99} = \frac{90}{9900} = 0.00909$$

Question 1

Suppose you randomly select a card from a standard deck of 52 cards. Let event A be 'the card is a heart' and event B be 'the card is a king.' What is the probability that the card is a heart or a king?

Accepted Answer

$$P(A 	ext{ or } B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52}$$

Question 2

If the probability that a randomly selected person has sleepwalked is 0.292, what is the probability that a randomly selected person has NOT sleepwalked?

Accepted Answer

$$1 - 0.292 = 0.708$$

Question 3

Two events A and B are said to be disjoint (mutually exclusive) if:

Accepted Answer

They cannot occur at the same time.

Question 4

A medical test for a disease has a 5% false positive rate. If two people are tested independently, what is the probability that both will have a false positive result?

Accepted Answer

$$0.05 	imes 0.05 = 0.0025$$

Question 5

Which of the following is the correct formula for the probability of the union of two events A and B?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) - P(A 	ext{ and } B)$$

Question 6

A bag contains 6 red balls and 4 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

Accepted Answer

$$\frac{6}{10} 	imes \frac{5}{9} = \frac{30}{90} = \frac{1}{3}$$

Question 7

In a population of 1,000,000 people, 2% have a certain genetic trait. If 3 people are randomly selected without replacement, what is the probability all three have the trait? (Use the 5% guideline for cumbersome calculations.)

Accepted Answer

$$(0.02)^3 = 0.000008$

Question 8

Which of the following best describes independent events?

Accepted Answer

The occurrence of one event does not affect the probability of the other.

Question 9

A factory uses two machines to produce widgets. Machine A produces 60% of the widgets and has a defect rate of 2%. Machine B produces 40% of the widgets and has a defect rate of 5%. What is the probability that a randomly selected widget is defective?

Accepted Answer

$$0.60 	imes 0.02 + 0.40 	imes 0.05 = 0.012 + 0.02 = 0.032$$

Question 10

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.2$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.4 + 0.5 - 0.2 = 0.7$$

Question 11

A new medication is tested on 100 patients. 30 experience side effects. If two patients are selected at random with replacement, what is the probability both experience side effects?

Accepted Answer

$$\frac{30}{100} 	imes \frac{30}{100} = 0.09$$

Question 12

Which of the following is an example of a compound event?

Accepted Answer

Rolling a die and getting an even number or a number greater than 4.

Question 13

A system has two independent backup generators, each with a 0.01 probability of failure during a power outage. What is the probability that both fail during the same outage?

Accepted Answer

$$0.01 	imes 0.01 = 0.0001$$

Question 14

If $$P(A) = 0.7$$, what is $$P(	ext{not } A)$$?

Accepted Answer

$$1 - 0.7 = 0.3$$

Question 15

A flight control system has three independent hydraulic systems, each with a 0.002 probability of failure on a flight. What is the probability that all three fail on a flight?

Accepted Answer

$$(0.002)^3 = 0.000000008$

Question 16

A survey finds that 40% of people like tea, 50% like coffee, and 20% like both. What is the probability that a randomly selected person likes tea or coffee?

Accepted Answer

$$0.4 + 0.5 - 0.2 = 0.7$$

Question 17

If two events are dependent, which of the following is true?

Accepted Answer

The occurrence of one event affects the probability of the other.

Question 18

A quality control inspector selects two light bulbs at random from a box of 100 bulbs, 10 of which are defective. What is the probability that both bulbs are defective if the selection is without replacement?

Accepted Answer

$$\frac{10}{100} 	imes \frac{9}{99} = \frac{90}{9900} = 0.00909$$

Probability: Addition and Multiplication Rules in Elementary Statistics

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