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...
9. An event A will occur with probability 0.4. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.24. We may conclude
You're right that these events are not independent, meaning that the probability of event B occurring depends on whether event A has occurred.
To find the overall probability of event B occurring, we need to consider the joint probability of events A and B occurring.
First, let's calculate the probability of event A occurring (P(A)) which is given as 0.9.
Next, we need to consider the probability of event B occurring given that event A has occurred. This is called the conditional probability and is denoted by P(B|A). In this case, P(B|A) is given as 0.2.
Using the formula for conditional probability, we multiply the probability of event A occurring by the conditional probability of event B given A:
P(A and B) = P(A) * P(B|A)
P(A and B) = 0.9 * 0.2
P(A and B) = 0.18
Finally, to find the overall probability of event B occurring, we can use the formula for marginal probability:
P(B) = P(A and B) + P(not A and B)
Since events A and B are mutually exclusive (meaning they cannot occur together), P(not A and B) = 0.
P(B) = P(A and B) + P(not A and B)
P(B) = 0.18 + 0
P(B) = 0.18
Therefore, the overall probability that event B will occur is 0.18 or 18%.