You deliver your friend a letter to be sent to your girlfriend, to be put in the mail. However, he can forget to put in the mail with probability $0.1$. If he doesn't forget, the probability that the delivery service don't deliver it is $0.1$. Finally, if the letter got sent, the probability that the girlfriend don't receive it is $0.1$
a) If your girlfriend didn't receive the letter, which is the probability that your friend forgot to put it in the mail?
The exercise is asking $P(\mbox{friend forgot | girlfriend didn't receive})$. I'd write this, using bayes theorem, as:
$P(\mbox{friend forgot | girlfriend didn't receive}) = \frac{P(\mbox{girlfriend didn't receive | friend forgot})P(\mbox{friend forgot})}{P(\mbox{something})}$
But I don't know what this something is.