5.3 Probability AP STATS

If the probability of event A is 0.3 and the probability of event B given A is 0.5, what is the probability of both A and B occurring?

• 0.15

• 0.20

• 0.25

• 0.30

Answer explanation

To find the probability of both A and B occurring, use the formula P(A and B) = P(A) * P(B|A). Here, P(A) = 0.3 and P(B|A) = 0.5. Thus, P(A and B) = 0.3 * 0.5 = 0.15, which is the correct answer.

• 0.1

• 0.2

• 0.3

• 0.4

Answer explanation

Since events A and B are independent, we can find P(A ∩ B) using the formula P(A ∩ B) = P(A) * P(B). Thus, P(A ∩ B) = 0.4 * 0.5 = 0.2. Therefore, the correct answer is 0.2.

Which of the following is true if two events, A and B, are independent?

Answer explanation

If two events A and B are independent, the probability of both events occurring together is the product of their individual probabilities: P(A ∩ B) = P(A) · P(B). Thus, the correct choice is P(A ∩ B) = P(A) · P(B).

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

Answer explanation

To find the probability of drawing 2 red balls, calculate the total ways to choose 2 balls from 5 (10 ways) and the ways to choose 2 red from 3 (3 ways). Thus, P(both red) = 3/10.

In a deck of 52 cards, what is the probability of drawing an Ace and then a King in succession without replacement?

Answer explanation

To find the probability of drawing an Ace and then a King without replacement, calculate: P(Ace) = 4/52, then P(King | Ace) = 4/51. Thus, P(Ace and King) = (4/52) * (4/51) = 16/2652 = 1/221.

• Yes

• No

Answer explanation

To check if A and B are independent, we verify if P(A ∩ B) = P(A) * P(B). Here, 0.24 = 0.6 * 0.4 = 0.24 holds true. Thus, A and B are independent events. The correct answer is Yes.

Using a tree diagram, if the probability of event A is 0.7 and the probability of event B given A is 0.5, what is the probability of event B?

• 0.35

• 0.50

• 0.70

• 0.85

Answer explanation

To find the probability of event B, use the formula P(B) = P(A) * P(B|A). Here, P(A) = 0.7 and P(B|A) = 0.5. Thus, P(B) = 0.7 * 0.5 = 0.35. Therefore, the correct answer is 0.35.

• 0.1

• 0.2

• 0.3

• 0.4

Answer explanation

To find P(A ∩ B), use the formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Plugging in the values: 0.7 = 0.5 + 0.4 - P(A ∩ B). This simplifies to P(A ∩ B) = 0.5 + 0.4 - 0.7 = 0.2.

What is the probability of rolling a 3 on a fair six-sided die and then flipping a head on a fair coin?

Answer explanation

The probability of rolling a 3 on a six-sided die is \(\frac{1}{6}\), and the probability of flipping a head on a coin is \(\frac{1}{2}\). The combined probability is \(\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}\).

• 0.1

• 0.2

• 0.4

• 0.8

Answer explanation

To find P(A ∩ B), use the formula P(A|B) = P(A ∩ B) / P(B). Rearranging gives P(A ∩ B) = P(A|B) * P(B) = 0.8 * 0.5 = 0.4. Thus, the correct answer is 0.4.