FM 12.6 Compound Probability

Two dice are rolled. What is the probability of getting a sum of 7 or 11?

Answer explanation

To find the probability of rolling a sum of 7 or 11, we count the successful outcomes. For 7, there are 6 combinations (1+6, 2+5, 3+4, etc.), and for 11, there are 2 (5+6, 6+5). Total successful outcomes = 8. Total outcomes = 36. Probability = 8/36 = 1/6.

A card is drawn from a standard deck of 52 cards. What is the probability of drawing a heart or a king?

Answer explanation

There are 13 hearts and 4 kings in a deck. However, one king is also a heart, so we count it only once. Thus, the total is 13 + 4 - 1 = 16 favorable outcomes. The probability is \frac{16}{52}

Which of the following pairs of events are mutually exclusive?

• Drawing a red card and drawing a king from a deck of cards

• Rolling a 3 and rolling an even number on a single die

• Drawing a queen and drawing a face card from a deck of cards

• Rolling a 5 and rolling a number greater than 2 on a single die

Answer explanation

Rolling a 3 and rolling an even number are mutually exclusive because a single die cannot show both a 3 (odd) and an even number at the same time.

A bag contains 3 red balls and 5 blue balls. Two balls are drawn one after the other without replacement. What is the probability that both balls are red?

Answer explanation

To find the probability of drawing 2 red balls without replacement, calculate: P(Red 1) = 3/8 and P(Red 2 | Red 1) = 2/7. Thus, P(both red) = (3/8) * (2/7) = 6/56 = 3/28. The correct answer is \frac{3}{28}.

Which of the following pairs of events are independent?

• Tossing a coin and rolling a die

• Drawing two cards from a deck without replacement

• Drawing a card and then drawing another card without replacement

• Choosing a marble from a bag and then choosing another without replacement

Answer explanation

Tossing a coin and rolling a die are independent events because the outcome of one does not affect the outcome of the other. In contrast, drawing cards or marbles without replacement affects the probabilities of subsequent draws.

What is the probability of drawing an ace or a spade from a standard deck of 52 cards?

Answer explanation

To find the probability of drawing an ace or a spade, we count 4 aces and 13 spades, but the ace of spades is counted twice. Thus, the total is 4 + 13 - 1 = 16. The probability is 16/52, which simplifies to 4/13.

A coin is tossed and a die is rolled. What is the probability of getting a head and a 4?

Answer explanation

The probability of getting a head when tossing a coin is \frac{1}{2}, and the probability of rolling a 4 on a die is \frac{1}{6}. Therefore, the combined probability is \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}.

Two cards are drawn from a standard deck of 52 cards with replacement. What is the probability that both cards are queens?

Answer explanation

The probability of drawing a queen is \frac{4}{52} = \frac{1}{13}. Since the draws are with replacement, the probability for both cards is \left(\frac{1}{13}\right)^2 = \frac{1}{169}. Thus, the correct answer is \frac{1}{169}.

Which of the following events are not mutually exclusive?

• Drawing a red card and drawing a black card from a deck of cards

• Rolling a 2 and rolling an even number on a single die

• Drawing a heart and drawing a diamond from a deck of cards

• Rolling a 1 and rolling a 6 on a single die

Answer explanation

Rolling a 2 and rolling an even number are not mutually exclusive because rolling a 2 is a specific case of rolling an even number. The other options represent events that cannot occur simultaneously.

A jar contains 4 red, 3 green, and 5 blue marbles. If two marbles are drawn at random with replacement, what is the probability that both are green?

Answer explanation

The probability of drawing a green marble is \frac{3}{12} = \frac{1}{4}. Since the draws are with replacement, the probability of both being green is \left(\frac{1}{4}\right)^2 = \frac{1}{16}. However, \frac{9}{144} simplifies to \frac{1}{16}, making it the correct choice.