$(5)[40$ marks$]$ In a movie, a gangster places two bullets in uniformly random chambers of the six$-$bullet cylinder of his revolver. He gives the cylinder a spin and says "Feeling lucky?" as he holds the gun against the hero's heart. Note that the cylinder moves clockwise by one chamber each time the trigger is pulled.

$-$ What is the probability that the hero will get shot the first time the gangster pulls the trigger?

marks$]$

$-$ Suppose the gangster pulls the trigger and the hero does not get shot. What is the probability that the hero will get shot if the gangster pulls the trigger a second time? $[3$ marks$]$

$-$ Even after the gangster pulls the trigger for the second time, the hero does not get shot. Irritated that luck is not in his favour, the gangster spins the cylinder again and pulls the trigger. Assuming that spinning the cylinder completely randomizes the positions of the chambers, calculate the probability that the hero will get shot on this third attempt? $[5$ marks$]$

This being a movie, the hero survives again! Mad with rage, the gangster spins the cylinder $($again completely randomizing the positions of the chambers$)$ and starts pulling the trigger one after the other. Let $\backslash ($ T $\backslash )$ be the r$.$v$.$ equal to the number of times the gangster has to pull the trigger until the hero gets shot.

$-$ For $\backslash ($ i $\backslash$in$[1,4]\backslash ),$ prove that $\backslash (\backslash$operatorname$\{$Pr$\}[$T$>$i$+1\backslash$mid T$>$i$]=\backslash$frac$\{4-$i$\}\{6-$i$\}[10\backslash )$ marks $\backslash (]\backslash )$

$-$ Using the above result and induction on $\backslash ($ i $\backslash ),$ prove that for $\backslash ($ i $\backslash$in$[1,5],\backslash$operatorname$\{$Pr$\}[$T$>$i$]=\backslash$frac$\{(6-$i$)(5-$i$)\}\{30\}[10\backslash )$ marks $\backslash (]\backslash )$

$-$ Using the above results, completely specify the PDF of $\backslash ($ T $\backslash )[10$ marks$]$