2020 VJC P2 Q7

In a game at a funfair, a player repeatedly tosses a fair six-sided die. If the toss is an odd number, the player loses and the game ends. Otherwise, the player continues to toss the die until a ‘\(6\)’ is shown and the game ends. The score, \(X\), is the number of tosses that it takes for the player to toss the first ‘\(6\)’, and is \(0\) if the player loses the game.

1. Show that \(\mathrm{P}\left( X=0 \right)=\frac{3}{4}\).[2]
2. What is the probability that the score is \(1\) given that the player does not lose?[2]

It is known that for \(0 < p < 1\), \(\sum\limits_{r=1}^{\infty }{r{{p}^{r-1}}=\frac{1}{{{\left( 1-p \right)}^{2}}}}\) and \(\sum\limits_{r=1}^{\infty }{{{r}^{2}}{{p}^{r-1}}=\frac{1+p}{{{\left( 1-p \right)}^{3}}}}\).

3. Show that \(\mathrm{E}\left( X \right)=\frac{3}{8}\). Find \(\mathrm{Var}\,(X)\).[4]