Parcels from sender S receiver R pass sequentially through two post-offices. Each post-office has a probability $\frac{1}{5}$ of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post-office is .

1. 0.444

The correct option is A 0.444
Let $E_1$:Parcel lost at Post Office - 1
Then $P\left(E_1\right)=\frac{1}{5}$

$E_2$:Parcel lost at Post Office - 2
$ThenP\left(E_2\right)=P\left({\overline{E}}_1\right)P\left(E_2\right)=\frac{4}{5}\times \frac{1}{5}=\frac{4}{25}$
A: Parcel lost

$ThenP\left(A\right)=P(E_1\cup E_2)=P\left(E_1\right)+P\left(E_2\right)$

$\text{(}as{E}_1,E_2\text{are mutually exclusive})$

$=\frac{1}{5}+\frac{4}{25}=\frac{9}{25}$


Required Probability = P(Lost by 2nd/Parcel is lost)

$=P\left(E_2/A\right)=\frac{P(E_2\cap A)}{P\left(A\right)}=\frac{P\left(E_2\right)}{P\left(A\right)}$

$=\frac{4/25}{9/25}=\frac{4}{9}=0.444$