Let's denote the following events:
- $ R $: It will rain.
- $ D $: The flight will be delayed.

We are given:
- $ P(R) = 0.1 $
- $ P(D) = 0.16 $
- $ P(R \cap D) = 0.02 $

We need to find the probability that it is not raining and the flight leaves on time. This can be expressed as $ P(\neg R \cap \neg D) $.

First, we use the formula for the complement of an event:
$$ P(\neg R) = 1 - P(R) $$
$$ P(\neg R) = 1 - 0.1 = 0.9 $$

Similarly,
$$ P(\neg D) = 1 - P(D) $$
$$ P(\neg D) = 1 - 0.16 = 0.84 $$

Next, we use the principle of inclusion-exclusion for the union of two events:
$$ P(R \cup D) = P(R) + P(D) - P(R \cap D) $$
$$ P(R \cup D) = 0.1 + 0.16 - 0.02 = 0.24 $$

The probability that it is not raining and the flight leaves on time is the complement of the union of $ R $ and $ D $:
$$ P(\neg R \cap \neg D) = 1 - P(R \cup D) $$
$$ P(\neg R \cap \neg D) = 1 - 0.24 = 0.76 $$

Therefore, the probability that it is not raining and the flight leaves on time is $ 0.76 $. Rounding to the nearest thousandth, the answer is $ 0.760 $.

Question

Let's denote the following events:
- $ R $: It will rain.
- $ D $: The flight will be delayed.

We are given:
- $ P(R) = 0.1 $
- $ P(D) = 0.16 $
- $ P(R \cap D) = 0.02 $

We need to find the probability that it is not raining and the flight leaves on time. This can be expressed as $ P(\neg R \cap \neg D) $.

First, we use the formula for the complement of an event:
$$ P(\neg R) = 1 - P(R) $$
$$ P(\neg R) = 1 - 0.1 = 0.9 $$

Similarly,
$$ P(\neg D) = 1 - P(D) $$
$$ P(\neg D) = 1 - 0.16 = 0.84 $$

Next, we use the principle of inclusion-exclusion for the union of two events:
$$ P(R \cup D) = P(R) + P(D) - P(R \cap D) $$
$$ P(R \cup D) = 0.1 + 0.16 - 0.02 = 0.24 $$

The probability that it is not raining and the flight leaves on time is the complement of the union of $ R $ and $ D $:
$$ P(\neg R \cap \neg D) = 1 - P(R \cup D) $$
$$ P(\neg R \cap \neg D) = 1 - 0.24 = 0.76 $$

Therefore, the probability that it is not raining and the flight leaves on time is $ 0.76 $. Rounding to the nearest thousandth, the answer is $ 0.760 $.

Knowee AI · Accepted Answer