Question

The probability that a student entering a university will graduate is $0.4$. Find the following probabilities out of $3$ students of the university:

1. none will graduate

2. only one will graduate

3. all will graduate.

Answer 1

Probability that $\alpha$ student graduates$=0.4$

(i) Probability that none of $3$ students graduate$=(1-0.4)\times (1-0.4)\times (1-0.4)$

Probability that student $\left(1\right)$ doesn't graduate

$\Rightarrow 0.6\times 0.6\times 0.6$

$\Rightarrow 0.216$

$\therefore$ Probability that none of $3$ students graduate$=0.216$

(ii) The only one will graduate.

we select one of $3$ in ${}^3{C}_1$ ways

his probability of passing $=0.4$

remaining two fail$\Rightarrow 0.6\times 0.6$

probability $={}^3{C}_1\times \left(0.4\right)\times \left(0.6\right)\times \left(0.6\right)$

$=3\times 0.4\times 0.36$

Probability that only one will graduate$\Rightarrow 0.432$

(iii) Probability that all students graduate$=\left(0.4\right)\left(0.4\right)\left(0.4\right)$

$=0.064$

$\therefore$ Probability that all will graduate$=0.064$.