<p>A bag contains $6$ counters labelled ‘R’ and $3$ counters labelled ‘G’. In a game, a player draws counters at random from the bag, one at a time with replacement, until $4$ counters have been drawn. Let $X$ be the number of times a counter labelled ‘R’ is drawn immediately after a counter labelled ‘G’ is drawn. For example, the sequences ‘RGGR’, ‘GGRR’ or ‘GRRR’ will give $X=1$ and the sequence ‘GRGR’ will give $X=2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\mathrm{P}(X=1)=\frac{46}{81}$ and tabulate the probability distribution of $X$.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The player loses $\$2$ if he does not have a counter labelled ‘R’ drawn immediately after a counter labelled ‘G’ is drawn out of the $4$ draws. Otherwise, the player wins $\$1.50x$ for $x$ number of times a counter labelled ‘R’ is drawn immediately after a counter labelled ‘G’ is drawn. Alvin and Bernard each plays one game. Find the probability that their total winnings is at least $\$4$.</p><p style="text-align: right;">[3]</p>

A bag contains $6$ counters labelled ‘R’ and $3$ counters labelled ‘G’. In a game, a player draws counters at random from the bag, one at a time with replacement, until $4$ counters have been drawn. Let $X$ be the number of times a counter labelled ‘R’ is drawn immediately after a counter labelled ‘G’ is drawn. For example, the sequences ‘RGGR’, ‘GGRR’ or ‘GRRR’ will give $X=1$ and the sequence ‘GRGR’ will give $X=2$.

Show that $\mathrm{P}(X=1)=\frac{46}{81}$ and tabulate the probability distribution of $X$.

Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

The player loses $\$2$ if he does not have a counter labelled ‘R’ drawn immediately after a counter labelled ‘G’ is drawn out of the $4$ draws. Otherwise, the player wins $\$1.50x$ for $x$ number of times a counter labelled ‘R’ is drawn immediately after a counter labelled ‘G’ is drawn. Alvin and Bernard each plays one game. Find the probability that their total winnings is at least $\$4$.

2018 NYJC MYE P2 Q7

2018 NYJC MYE P2 Q7

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