<p>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.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">What is the probability that the score is $1$ given that the player does not lose?</div><p style="text-align: right;">[2]</p></li></ol>
<p>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 $\sum\limits_{r=1}^{\infty }{{{r}^{2}}{{p}^{r-1}}=\frac{1+p}{{{\left( 1-p \right)}^{3}}}}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that $\mathrm{E}\left( X \right)=\frac{3}{8}$. Find $\mathrm{Var}\,(X)$.</div><p style="text-align: right;">[4]</p></li></ol>

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.

Show that $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$.

What is the probability that the score is $1$ given that the player does not lose?

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 $\sum\limits_{r=1}^{\infty }{{{r}^{2}}{{p}^{r-1}}=\frac{1+p}{{{\left( 1-p \right)}^{3}}}}$.

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

Discrete Random Variables (DRV) - General

Discrete Random Variables (DRV)

H2 Math

International Baccalaureate Higher Level Mathematics

2020 VJC P2 Q7

2020 VJC P2 Q7

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