Question

1. identify the following pairs as mutually exclusive or not. explain your reasoning.

a. rolling a pair of dice: getting a sum of 6; getting one die with a 6 on it

b. flipping a coin 7 times: getting exactly 3 heads; getting exactly 5 heads

c. flipping a coin 7 times: getting at least 3 heads; getting at least 5 heads

Explanation:

Step1: Recall definition of mutually - exclusive

Two events are mutually exclusive if they cannot occur at the same time.

Step2: Analyze part a

The possible combinations for a sum of 6 when rolling two dice are (1,5), (2,4), (3,3), (4,2), (5,1). The combinations with a 6 on one die are (1,6), (2,6), (3,6), (4,6), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6). There are no common combinations. So they are mutually exclusive.

Step3: Analyze part b

When flipping a coin 7 times, getting exactly 3 heads means there are 3 heads and 4 tails, and getting exactly 5 heads means there are 5 heads and 2 tails. These two events cannot happen simultaneously. So they are mutually exclusive.

Step4: Analyze part c

Getting at least 3 heads means 3, 4, 5, 6, or 7 heads. Getting at least 5 heads means 5, 6, or 7 heads. The cases of 5, 6, 7 heads are common to both events. So they are not mutually exclusive.

Answer:

a. Mutually exclusive. Reason: No common combinations for sum of 6 and having a 6 on one die.
b. Mutually exclusive. Reason: Cannot have exactly 3 heads and exactly 5 heads simultaneously when flipping a coin 7 times.
c. Not mutually exclusive. Reason: There are common cases (5, 6, 7 heads) for getting at least 3 heads and getting at least 5 heads.