In an examination, 60% of the candidates passed. If 10 candidates are selected at random, find the probability that; (1) at least two of the, failed (2) exa...
Question 1 Report
In an examination, 60% of the candidates passed. If 10 candidates are selected at random, find the probability that;
(1) at least two of the, failed
(2) exactly half of them passed
(3) at most two of them failed
Answer Details
This is a binomial situation. Let \(X\) be the number who fail. A candidate fails with probability \(p=0.4\) and passes with probability \(0.6\), with \(n=10\).
\[P(X=r)=\binom{10}{r}(0.4)^{r}(0.6)^{10-r}\]
(1) At least two failed: \(P(X\ge 2)\)
\[P(X\ge2)=1-P(0)-P(1)\]
\(P(0)=(0.6)^{10}=0.00605\) and \(P(1)=\binom{10}{1}(0.4)(0.6)^{9}=10(0.4)(0.010078)=0.04031\).
\[P(X\ge2)=1-0.00605-0.04031=0.954\]
(2) Exactly half passed means \(5\) passed and \(5\) failed, i.e. \(X=5\).
\[P(X=5)=\binom{10}{5}(0.4)^{5}(0.6)^{5}=252(0.01024)(0.07776)=0.201\]
(3) At most two failed: \(P(X\le 2)=P(0)+P(1)+P(2)\)
\(P(2)=\binom{10}{2}(0.4)^{2}(0.6)^{8}=45(0.16)(0.016796)=0.12093\).
\[P(X\le2)=0.00605+0.04031+0.12093=0.167\]
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