Events and are mutually exclusive. Suppose event occurs with probability and event occurs with probability .
Compute the probability that occurs or does not occur (or both).
Compute the probability that either occurs without occurring or and both occur.
To answer these questions, we need to use probability concepts:
1. Probability that either event A occurs or event B does not occur (or both):
To calculate this, we can use the principle of inclusion-exclusion. We start by calculating the probability of event A occurring and event B not occurring:
P(A and not B) = P(A) * (1 - P(B))
Next, we calculate the probability of event B occurring and event A not occurring:
P(not A and B) = P(B) * (1 - P(A))
Finally, the probability that either event A occurs or event B does not occur (or both) is the sum of the two probabilities calculated above:
P(A or not B) = P(A and not B) + P(not A and B)
2. Probability that either event A occurs without event B occurring or both event A and event B occur:
Here, we calculate the probability of event A occurring without event B occurring, and the probability of both events A and B occurring separately. Then, we sum them up.
P(A without B) = P(A) * (1 - P(B))
P(A and B) = P(A) * P(B)
Finally, we calculate the probability by summing the probabilities we just obtained:
P((A without B) or (A and B)) = P(A without B) + P(A and B)
Note: In both cases, it is essential to have the individual probabilities of events A and B. Please provide those probabilities, and I can help you calculate the final probabilities.