Question at position 3 Select a random integer from -100 to 100. Which of the following pairs of events are mutually exclusive (disjoint)?A: the number is positive; B: the number is oddA: the number is odd; B: the number is greater than 10A: the number is less than 55; B: the number is negativeA: the number is above 50; B: the number is less than 20A: the number is odd; B: the number is 13

Question

Question at position 3 Select a random integer from -100 to 100. Which of the following pairs of events are mutually exclusive (disjoint)?A: the number is positive; B: the number is oddA: the number is odd; B: the number is greater than 10A: the number is less than 55; B: the number is negativeA: the number is above 50; B: the number is less than 20A: the number is odd; B: the number is 13

BlackTom AI · Accepted Answer

Consider the concept of mutually exclusive (disjoint) events: two events are disjoint if there is no possible outcome that makes both events occur at the same time.

Option 1: A: the number is positive; B: the number is odd. Since there exist positive odd numbers (e.g., 1, 3, 5, ...), both A and B can occur simultaneously. Therefore, these events are not disjoint.

Option 2: A: the number is odd; B: the number is greater than 10. There are numbers that are both odd and greater than 10 (e.g., 11, 13). Hence, these events are not disjoint.

Option 3: A: the number is less than 55; B: the number is negative. Negative numbers are all less than 55, so there are outcomes that satisfy both A and B (e.g., -3). Therefore, these events are not disjoint.

Option 4: A: the number is above 50; B: the number is less than 20. Any number above 50 cannot simultaneously be less than 20. There is no overlap between these two conditions within the given range (-100 to 100). Thus, these events are disjoint.

Option 5: A: the number is odd; B: the number is 13. The number 13 is odd, so there exists at least one outcome that makes both events occur. Therefore, these events are not disjoint.

In summary, only Option 4 yields disjoint events, since the ranges (above 50) and (less than 20) do not intersect within the specified sample space.

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