Rain and the bus 3. To decide whether the rain and the bus running late are dependent or independent events, first define the two events and then write their probabilities as decimals. (3 points) Let event A = . P(A) = . Let event B = . P(B) = . A and B = . P(A and B) = . 4. Use the probabilities from question 3 to decide whether the rain and the bus running late are independent or dependent events. (2 points: 1 point for the correct math, 1 point for the conclusion)

The event that it rains and the event that the bus is late aren't independent events, rather, they are very much dependent on each other.

Step-by-step explanation:

- Let event A be the event that it rains.

P(A) = 0.20

- Let event B be the event that the bus is late

P(B) = 0.08

- Let the event that it rains and the bus is late be A and B, that is, (A n B)

From the question, P(A n B) = 0.03

Two events are said to be independent when the probability of one of the events happening does not in any way affect the probability of the other event happening, and vice versa.

Mathematically, independent events are related through the expression

P(A n B) = P(A) × P(B)

And the conditional expression P(A|B) or P(B|A) for independent events are given as

P(A|B) = P(A) And P(B|A) = P(B)

So, we check each of the mathematical proofs.

P(A) = 0.20, P(B) = 0.08, P(A n B) = 0.03

P(A) × P(B) = 0.20 × 0.08 = 0.016 ≠ 0.03

Hence, the two events aren't independent.

Further checks,

The conditional probabilities P(A|B) and P(B|A) are normally given as

P(A|B) = P(A n B) ÷ P(B)

P(A|B) = 0.03 ÷ 0.08 = 0.375 ≠ P(A) (which is 0.2)

Further proof that they are dependent events.

P(B|A) = P(A n B) ÷ P(A)

P(B|A) = 0.03 ÷ 0.20 = 0.15 ≠ P(B) (which is 0.08)

Further proof that the two events aren't independent.

Hope this Helps!!!