Q: A spinner is divided into 4 equal regions: red, blue, yellow, and green. What is the probability of spinning yellow on the first spin and NOT yellow on the second spin?

$$\dfrac{1}{4} \times \dfrac{3}{4} = \dfrac{3}{16}$$

Q: Which formula correctly calculates the probability of two independent events A and B both occurring?

$$P(A \cap B) = P(A) \times P(B)$$

Q: A fair coin is flipped twice. What is the probability of getting heads on both flips?

$$\dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}$$

Q: A die is rolled twice. What is the probability of rolling an even number on the first roll and a 3 on the second roll?

$$\dfrac{3}{6} \times \dfrac{1}{6} = \dfrac{1}{12}$$

Q: If three independent events have probabilities $$P(A) = 0.5$$, $$P(B) = 0.2$$, and $$P(C) = 0.8$$, what is the probability that all three occur?

$$0.5 \times 0.2 \times 0.8 = 0.08$$

Q: A spinner with 4 equal regions numbered 1-4 is spun three times. What is the probability of landing on yellow on the first spin, an even number on the second spin, and blue or red on the third spin?

$$\dfrac{1}{4} \times \dfrac{2}{4} \times \dfrac{2}{4} = \dfrac{1}{16}$$

Q: A bag contains 5 red and 5 blue marbles. If you draw one marble, replace it, and draw again, what is the probability of drawing a red marble both times?

$$\dfrac{5}{10} \times \dfrac{5}{10} = \dfrac{1}{4}$$

Q: Suppose you flip a coin and roll a die. What is the probability of getting tails and rolling a 6?

$$\dfrac{1}{2} \times \dfrac{1}{6} = \dfrac{1}{12}$$

Question 1

Which of the following scenarios best illustrates independent events?

Accepted Answer

Rolling a die and flipping a coin.

Question 2

If $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and A and B are independent events, what is $$P(A \cap B)$$?

Accepted Answer

$$0.2$$

Question 3

A spinner is divided into 4 equal regions: red, blue, yellow, and green. What is the probability of spinning yellow on the first spin and NOT yellow on the second spin?

Accepted Answer

$$\dfrac{1}{4} 	imes \dfrac{3}{4} = \dfrac{3}{16}$$

Question 4

Which formula correctly calculates the probability of two independent events A and B both occurring?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B)$$

Question 5

A fair coin is flipped twice. What is the probability of getting heads on both flips?

Accepted Answer

$$\dfrac{1}{2} 	imes \dfrac{1}{2} = \dfrac{1}{4}$$

Question 6

A die is rolled twice. What is the probability of rolling an even number on the first roll and a 3 on the second roll?

Accepted Answer

$$\dfrac{3}{6} 	imes \dfrac{1}{6} = \dfrac{1}{12}$$

Question 7

If three independent events have probabilities $$P(A) = 0.5$$, $$P(B) = 0.2$$, and $$P(C) = 0.8$$, what is the probability that all three occur?

Accepted Answer

$$0.5 	imes 0.2 	imes 0.8 = 0.08$$

Question 8

A spinner with 4 equal regions numbered 1-4 is spun three times. What is the probability of landing on yellow on the first spin, an even number on the second spin, and blue or red on the third spin?

Accepted Answer

$$\dfrac{1}{4} 	imes \dfrac{2}{4} 	imes \dfrac{2}{4} = \dfrac{1}{16}$$

Question 9

Which of the following statements is TRUE about independent events?

Accepted Answer

The probability of both events occurring is the product of their probabilities.

Question 10

A bag contains 5 red and 5 blue marbles. If you draw one marble, replace it, and draw again, what is the probability of drawing a red marble both times?

Accepted Answer

$$\dfrac{5}{10} 	imes \dfrac{5}{10} = \dfrac{1}{4}$$

Question 11

Suppose you flip a coin and roll a die. What is the probability of getting tails and rolling a 6?

Accepted Answer

$$\dfrac{1}{2} 	imes \dfrac{1}{6} = \dfrac{1}{12}$$

Question 12

If $$P(A) = 0.7$$ and $$P(B) = 0.3$$, and A and B are independent, what is the probability that at least one occurs?

Accepted Answer

$$1 - (1 - 0.7)(1 - 0.3) = 0.79$$

Question 13

Which of the following is NOT a property of independent events?

Accepted Answer

$$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} = P(B)$$

Question 14

A medical test is positive with probability 0.95 if a patient has a disease, and the probability that a randomly selected patient has the disease is 0.01. What is the probability that a randomly selected patient both has the disease and tests positive?

Accepted Answer

$$0.95 	imes 0.01 = 0.0095$$

Question 15

If you roll a die three times, what is the probability that you get a 2 on the first roll, a 4 on the second, and a 6 on the third?

Accepted Answer

$$\dfrac{1}{6} 	imes \dfrac{1}{6} 	imes \dfrac{1}{6} = \dfrac{1}{216}$$

Question 16

A bag contains 3 green, 2 blue, and 5 red marbles. If you draw one marble, replace it, and draw again, what is the probability of drawing a green marble first and a blue marble second?

Accepted Answer

$$\dfrac{3}{10} 	imes \dfrac{2}{10} = \dfrac{6}{100} = 0.06$$

Question 17

Which of the following is the correct general formula for the probability of $$n$$ independent events all occurring?

Accepted Answer

$$P(A_1$ \cap A_2$ \cap \ldots \cap A_n) = P(A_1$) 	imes P(A_2$) 	imes \ldots 	imes P(A_n)$$

Question 18

A fair coin is flipped three times. What is the probability of getting heads on the first flip, tails on the second, and heads on the third?

Accepted Answer

$$\dfrac{1}{2} 	imes \dfrac{1}{2} 	imes \dfrac{1}{2} = \dfrac{1}{8}$$

Question 19

A spinner with 4 equal regions is spun twice. What is the probability of landing on blue both times?

Accepted Answer

$$\dfrac{1}{4} 	imes \dfrac{1}{4} = \dfrac{1}{16}$$

Probability of Multiple Independent Events: Multiplication Rule

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