In a university examination, which was indeed very tough, 50% at jeast failed in "Statistics",75%...

Given
P(failed in Statistics) ${\displaystyle =0.50}$
P(failed in Topology) ${\displaystyle =0.75}$
P(failed in Functional Analysis) ${\displaystyle =0.82}$
P(failed in Applied Maths) ${\displaystyle =0.96}$
To find:
How many failed in at least all four
A, B, C and D are said to be independent when : ${\displaystyle P(A\cap B\cap C\cap D)=P\left(B\right)\times P\left(C\right)\times P\left(D\right)}$
The probability of failing in at least all four is given by:
P(failed in Statistics ${\displaystyle \cap }$ failed in Topology ${\displaystyle \cap }$ failed in Functional Analysis ${\displaystyle \cap }$ failed in Applied Maths)
= P(failed in Statistics) ${\displaystyle \times }$ P(failed in Topo log y) X P(failed in Functional Analysis) ${\displaystyle \times }$ P(failed in Applied Maths)
${\displaystyle [\because independentevents]}$
${\displaystyle =0.50\times 0.75\times 0.82\times 0.96}$
=0. 2952
${\displaystyle \stackrel{\sim }{=}0.3}$
Therefore, ${\displaystyle 3\mathrm{\%}}$ of them at least failed in all the four.