Two risks are independent of each other.

The probability of each event is unknown; but we can estimate the probability of each event.

We estimated a range, for each event, of 2% to 3%.

What is the range of probability of them both occurring?

Question

Two risks are independent of each other.

The probability of each event is unknown; but we can estimate the probability of each event.

We estimated a range, for each event, of 2% to 3%.

What is the range of probability of them both occurring?

BlackTom AI · Accepted Answer

We’re told two risks are independent and that each event has an estimated probability between 2% and 3%. To find the range for them both occurring, we multiply the probabilities because independence implies P(A and B) = P(A) × P(B).

First, consider the smallest possible product: if both events are at the lower end, 2% is 0.02 in decimal terms. 0.02 × 0.02 = 0.0004, which is 0.04%.

Next, consider the largest possible product: if both events are at the upper end, 3% is 0.03 in decimal terms. 0.03 × 0.03 = 0.0009, which is 0.09%.

Thus, the probability of both events occurring, given independence and the 2%–3% estimate for each, lies between 0.04% and 0.09%.

Now evaluate the answer choices:
- Option 1: ".04% to .09%" matches the calculated range exactly, so this is consistent with the independence assumption and the given bounds.
- Option 2: "less than 20%" is far too large; even at the extremes, the probability is under 0.1%, not anywhere near 20%
- Option 3: "4% to 9%" confuses the decimal placement by a factor of 100; 0.04%–0.09% is not 4%–9%
- Option 4: "Greater than 30%" is completely incorrect given the tiny estimated probabilities and product rule

In summary, the correct interpretation for independent events with each probability between 2% and 3% yields a joint probability range of 0.04% to 0.09%, corresponding to Option 1.

Two risks are independent of each other. The probability of each event is unknown; but we can estimate the probability of each event. We estimated a range, for each event, of 2% to 3%. What is the range of probability of them both occurring?单项选择题

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