The probabilities that Ali, Baba and Katty will gain admission to college are \(\frac{2}{3}, \frac{3}{4}\) and \(\frac{4}{5}\) respectively. Find the probab...
Question 1 Report
The probabilities that Ali, Baba and Katty will gain admission to college are \(\frac{2}{3}, \frac{3}{4}\) and \(\frac{4}{5}\) respectively. Find the probability that:
(a) only Katty and Baba will gain admission ;
(b) none of them will gain admission ;
(c) at most two of them will gain admission.
Answer Details
Admission probabilities: Ali \(P(A)=\tfrac23\), Baba \(P(B)=\tfrac34\), Katty \(P(K)=\tfrac45\). Failures: \(P(A')=\tfrac13,\ P(B')=\tfrac14,\ P(K')=\tfrac15\). The three events are independent.
(a) Only Katty and Baba (so Ali fails):
\[P(A')\,P(B)\,P(K)=\tfrac13\times\tfrac34\times\tfrac45=\frac{12}{60}=\frac15\]
(b) None gains admission:
\[P(A')\,P(B')\,P(K')=\tfrac13\times\tfrac14\times\tfrac15=\frac{1}{60}\]
(c) At most two = not all three. So subtract the probability that all three gain admission:
\[1-P(A)\,P(B)\,P(K)=1-\left(\tfrac23\times\tfrac34\times\tfrac45\right)=1-\frac{24}{60}=1-\frac25=\frac35\]
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