Answer :
To solve this problem involving events A, B, and C, we need to understand how their probabilities interact based on the given conditions.
10.1.1 Both A and C occur:
Since A and C are mutually exclusive events, they cannot occur at the same time. Mutually exclusive means that if one event happens, the other cannot. Therefore, the probability of both A and C occurring is:
[tex]P(A \cap C) = 0[/tex]
10.1.2 Both B and C occur:
B and C are independent events. The probability that both B and C occur is found by multiplying their individual probabilities:
[tex]P(B \cap C) = P(B) \times P(C) = 0.4 \times 0.2 = 0.08[/tex]
10.1.3 At least ONE of A or B occurs:
To find the probability that at least one of A or B occurs, we use the formula for the probability of the union of two independent events:
[tex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/tex]
Since A and B are independent, the probability of both A and B occurring is:
[tex]P(A \cap B) = P(A) \times P(B) = 0.3 \times 0.4 = 0.12[/tex]
Now, substitute the values into the formula:
[tex]P(A \cup B) = 0.3 + 0.4 - 0.12 = 0.58[/tex]
Therefore, the probability of at least one of A or B occurring is 0.58.
In summary:
The probability of both A and C occurring is 0.
The probability of both B and C occurring is 0.08.
The probability of at least one of A or B occurring is 0.58.
