<p>There are three identically shaped balls, numbered from $1$ to $3$, in a bag. Balls are drawn one by one at random and with replacement. The random variable $X$ is the number of draws needed for any ball to be drawn a second time. The two draws of the same ball do not need to be consecutive.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\mathrm{P}\left( X=4 \right)=\frac{2}{9}$ and find the probability distribution of $X$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that $\mathrm{E}\left( X \right)=\frac{26}{9}$ and find the exact value of $\mathrm{Var}\left( X \right)$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">The mean for forty-four independent observations of $X$ is denoted by $\bar{X}$ . Using a suitable approximation, find the probability that $\bar{X}$ exceeds $3$.</span></div><p style="text-align: right;">[3]</p></li></ol>

There are three identically shaped balls, numbered from $1$ to $3$, in a bag. Balls are drawn one by one at random and with replacement. The random variable $X$ is the number of draws needed for any ball to be drawn a second time. The two draws of the same ball do not need to be consecutive.

Show that $\mathrm{P}\left( X=4 \right)=\frac{2}{9}$ and find the probability distribution of $X$.

Show that $\mathrm{E}\left( X \right)=\frac{26}{9}$ and find the exact value of $\mathrm{Var}\left( X \right)$.

The mean for forty-four independent observations of $X$ is denoted by $\bar{X}$ . Using a suitable approximation, find the probability that $\bar{X}$ exceeds $3$.

2017 NJC P2 Q7

2017 NJC P2 Q7

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