A building contractor tendered for two independent contracts, X and Y. The probabilities that he will win contract X is 0.5 and not win contract Y is 0.3, W...
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A building contractor tendered for two independent contracts, X and Y. The probabilities that he will win contract X is 0.5 and not win contract Y is 0.3, What is the probability that he will win :
(a) both contracts ;
(b) exactly one of the contracts ;
(c) neither of the contracts?
Answer Details
Setup. The contracts are independent. Given:
\[P(\text{win }X) = 0.5,\qquad P(\text{not win }Y) = 0.3 \Rightarrow P(\text{win }Y) = 0.7\]
Also \(P(\text{not win }X) = 0.5\).
(a) Wins both contracts
\[P(X\cap Y) = 0.5\times 0.7 = 0.35\]
(b) Wins exactly one contract
Either win \(X\) and lose \(Y\), or lose \(X\) and win \(Y\):
\[P = (0.5\times 0.3) + (0.5\times 0.7) = 0.15 + 0.35 = 0.50\]
(c) Wins neither contract
\[P(\text{not }X\cap\text{not }Y) = 0.5\times 0.3 = 0.15\]
Check: \(0.35 + 0.50 + 0.15 = 1.00\), confirming all outcomes are accounted for.
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