Let's solve the given problem step by step. We are given that $A$, $B$, and $C$ are mutually exclusive events, meaning they cannot occur simultaneously. The probabilities are:

• $P(A) = 0.23$
• $P(B) = 0.34$
• $P(C) = 0.41$

We need to find the following probabilities:

(a) $P(A \cup B \cup C)$

Since $A$, $B$, and $C$ are mutually exclusive, the probability of their union is the sum of their individual probabilities:

$P(A \cup B \cup C) = P(A) + P(B) + P(C)$

Substituting the given values:

$P(A \cup B \cup C) = 0.23 + 0.34 + 0.41 = 0.98$

(b) $P(A \cap B \cap C)$

Because $A$, $B$, and $C$ are mutually exclusive, they cannot occur together. Hence, the intersection of these events is 0:

$P(A \cap B \cap C) = 0$

(c) $P(A \cap B)$

For mutually exclusive events, the intersection between any two events is also 0:

$P(A \cap B) = 0$

(d) $P((A \cup B) \cap C)$

This is asking for the probability of the intersection of $C$ with the union of $A$ and $B$. Since $A$, $B$, and $C$ are mutually exclusive, the intersection of $C$ with $A \cup B$ is 0:

$P((A \cup B) \cap C) = 0$

(e) $P(A' \cap B' \cap C')$

This is the probability that none of the events $A$, $B$, or $C$ occur. To calculate this, we first find the complement of each event:

$P(A') = 1 - P(A) = 1 - 0.23 = 0.77$ $P(B') = 1 - P(B) = 1 - 0.34 = 0.66$ $P(C') = 1 - P(C) = 1 - 0.41 = 0.59$

Since $A$, $B$, and $C$ are mutually exclusive, their complements are independent. Thus, we can multiply the probabilities:

$P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') = 0.77 \times 0.66 \times 0.59 = 0.300$

Rounding to two decimal places:

$P(A' \cap B' \cap C') = 0.30$

Summary of Results:

• (a) $P(A \cup B \cup C) = 0.98$
• (b) $P(A \cap B \cap C) = 0$
• (c) $P(A \cap B) = 0$
• (d) $P((A \cup B) \cap C) = 0$
• (e) $P(A' \cap B' \cap C') = 0.30$

Would you like further clarification or details?

Here are five related questions:

1. How would the probabilities change if $A$, $B$, and $C$ were not mutually exclusive?
2. What is the general formula for $P(A \cup B)$ when events are not mutually exclusive?
3. How do you calculate the complement of the union of three events?
4. If $P(A)$, $P(B)$, and $P(C)$ were dependent, how would you approach calculating $P(A \cap B)$?
5. Can mutually exclusive events ever be independent?

Tip: Mutually exclusive events cannot occur at the same time, while independent events do not influence each other's occurrence.