Event A has probability 0.9 of occurring. Event B has 0.2 chance of occurring if, and only if, event A has occurred. What is the overall probability that event B will occur?

I'm so confused. This seems so simple, I'm not sure why I can't reason through it. They're not independent events...

You're right that the two events are not independent, which can make it a bit trickier to reason through. However, we can break down the problem step by step.

Let's start with event A. We know that the probability of event A occurring is 0.9, so we can represent this as P(A) = 0.9.

Now, we need to consider event B. We are given that event B has a 0.2 chance of occurring if, and only if, event A has occurred. This means that the probability of event B occurring depends on whether event A has occurred.

To calculate the overall probability of event B occurring, we need to take into account both the probability of event A occurring and the probability of event B occurring given that event A has occurred.

Mathematically, we can express this as P(B) = P(B|A) * P(A), where P(B) represents the overall probability of event B occurring, P(B|A) represents the probability of event B occurring given that event A has occurred, and P(A) represents the probability of event A occurring.

We are given that P(B|A) = 0.2, which is the probability of event B occurring given that event A has occurred.

Plugging in the values, we have P(B) = 0.2 * 0.9 = 0.18.

Therefore, the overall probability that event B will occur is 0.18 (or 18%).

To summarize the steps:
1. Calculate the probability of event A occurring: P(A) = 0.9.
2. Calculate the probability of event B occurring given that event A has occurred: P(B|A) = 0.2.
3. Multiply P(B|A) by P(A) to obtain the overall probability of event B occurring: P(B) = P(B|A) * P(A).
4. Calculate the value of P(B) to find the overall probability. In this case, P(B) = 0.2 * 0.9 = 0.18 (or 18%).