Events $$ A $$ and $$ B $$ are mutually exclusive. Suppose event $$ A $$ occurs with probability, 0.43 and event $$ B $$ occurs with probability 0.18 . Compute the following. (If necessary, consult a list of formulas.) (a) Compute the probability that $$ B $$ occurs but $$ A $$ does not occur. (b) Compute the probability that either $$ B $$ occurs without $$ A $$ occurring or $$ A $$ and $$ B $$ both occur. $$ \square $$

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UpStudy AI · Accepted Answer

Given that events $$ A $$ and $$ B $$ are **mutually exclusive**, they cannot occur simultaneously. This implies:
$$
P(A \cap B) = 0
$$
Additionally, we're provided with:
$$
P(A) = 0.43 \quad \text{and} \quad P(B) = 0.18
$$

Let's address each part of the problem:

### (a) Probability that $$ B $$ occurs but $$ A $$ does not occur.

Since $$ A $$ and $$ B $$ are mutually exclusive, if $$ B $$ occurs, $$ A $$ **cannot** occur. Therefore:
$$
P(B \text{ and not } A) = P(B) = 0.18
$$
**Answer:** $$ 0.18 $$

### (b) Probability that either $$ B $$ occurs without $$ A $$ occurring or $$ A $$ and $$ B $$ both occur.

This can be interpreted as:
$$
P\left( (B \text{ and not } A) \text{ or } (A \text{ and } B) \right)
$$
Breaking it down:
$$
P(B \text{ and not } A) + P(A \text{ and } B)
$$
Given that $$ A $$ and $$ B $$ are mutually exclusive:
$$
P(A \text{ and } B) = 0
$$
Thus:
$$
P(B \text{ and not } A) + 0 = 0.18 + 0 = 0.18
$$
**Answer:** $$ 0.18 $$
(a) The probability that $$ B $$ occurs but $$ A $$ does not occur is 0.18. (b) The probability that either $$ B $$ occurs without $$ A $$ occurring or both $$ A $$ and $$ B $$ occur is 0.18.

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