Calculating Team Win Probability with Captaincy
Problem Breakdown
This problem involves finding the overall probability of the team winning the tournament. We need to consider two distinct scenarios based on who is selected as captain: either player A is captain or player B is captain. These are mutually exclusive events.
We are given the following probabilities:
- Probability of A being captain: $P(A) = 0.6$
- Probability of B being captain: $P(B) = 0.4$
- Probability of winning given A is captain: $P(\text{Win}|A) = 0.8$
- Probability of winning given B is captain: $P(\text{Win}|B) = 0.7$
We need to find the total probability of the team winning, $P(\text{Win})$.
Total Probability Calculation
We can use the Law of Total Probability. The formula is:
$ P(\text{Win}) = P(\text{Win}|A) \times P(A) + P(\text{Win}|B) \times P(B) $
This formula sums the probabilities of winning through each possible captaincy scenario.
Step 1: Calculate probability of winning with A as captain
Multiply the probability of A being captain by the probability of winning if A is captain:
$ P(\text{Win} \cap A) = P(\text{Win}|A) \times P(A) = 0.8 \times 0.6 = 0.48 $
Step 2: Calculate probability of winning with B as captain
Multiply the probability of B being captain by the probability of winning if B is captain:
$ P(\text{Win} \cap B) = P(\text{Win}|B) \times P(B) = 0.7 \times 0.4 = 0.28 $
Step 3: Sum probabilities for the total win probability
Add the probabilities calculated in the previous steps:
$ P(\text{Win}) = 0.48 + 0.28 = 0.76 $
Conclusion
The total probability that the team wins the tournament is 0.76.
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at random is tossed and head turns up. If the probability that the drawn coin was unbiased,
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