Question

The probability that a person who undergoes kidney operation will recover is $0.5$. Find the probability that of the six patients who undergo similar operations. (a) None will recover. (b) Half of them will recover

Answer 1

Sine there are six patients, $n=6$.
If a patient recovers, the outcomes is a success.
Thus, $P=p$ (success) $=0.5$ and $q=1-p=0.5$
Let X : No. of patients who recovered out of $6$.
Then $X\sim b(n=6,p=0.5)$
The p.m.f. of $X$ is given as,
$P(X=x)=p\left(x\right){=}^6{C}_x\left(0.5{)}^x\right(0.3{)}^{6-x},x=0,1,2,3,4,5,6$.
(i) Probability (none will recover) $=P(X=0)$
${=}^6{C}_0\left(0.5{)}^0\right(0.5{)}^6$
$=\left(1\right)\cdot \left(1\right)\cdot (0.5{)}^6$
$=0.015625$
(ii) Probability (Half of them will recover) $=P(X=3)$
${=}^6{C}_2\left(0.5{)}^3\right(0.5{)}^3$
$=20\times 0.125\times 0.125$
$=0.3125$