91Ó°ÊÓ

Determine the probability distribution of the number of spades in a 5 card poker hand from an ordinary deck of 52 cards.

An absent-minded professor has 5 keys. One of the keys opens the door to his apartment. One night he arrives at his building, reaches into his pocket and selects a key at random from those on his chain. He tries it in the lock. If it doesn't work, he replaces the key and again selects at random from the 5 keys. He continues this process until he finally finds his key, then stops. Let \(\mathrm{X}=\) the number of attempts the professor makes. What is the probability distribution of \(\mathrm{X}\) ?

Let a distribution function \(\mathrm{F}\), be given by $$ \begin{aligned} \mathrm{F}(\mathrm{x}) &=0 ; & & \mathrm{x}<0 \\ &=[(\mathrm{x}+1) / 2] ; & & 0 \leq \mathrm{x}<1 \\ &=1 ; & & 1 \leq \mathrm{x} \end{aligned} $$ Find \(\operatorname{Pr}[-3<\mathrm{x} \leq(1 / 2)]\) and sketch \(\mathrm{F}\).