<p>The random variable \(S\) is the number of successes in \(5\) independent trials of an experiment in which the probability of success in any trial is \(\frac{1}{3}\). The random variable \(D\) is the difference between the number of successes and the number of failures in \(5\) such trials.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State the values that \(D\) can take.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that \(\mathrm{P}(D=1)=\frac{40}{81}\).</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By finding the probability distribution of \(D\), find the exact value of \(\mathrm{E}(D^2)\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the exact values of \(\mathrm{E}(S)\) and \(\mathrm{E}(S^2)\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, by showing that \(D^2 = 4S^2 - 20S + 25\), verify the correctness of the value of \(\mathrm{E}(D^2)\) found in part (c).</span><span class="inline-marks">[2]</span></div></li></ol>

The random variable \(S\) is the number of successes in \(5\) independent trials of an experiment in which the probability of success in any trial is \(\frac{1}{3}\). The random variable \(D\) is the difference between the number of successes and the number of failures in \(5\) such trials.

Show that \(\mathrm{P}(D=1)=\frac{40}{81}\).

By finding the probability distribution of \(D\), find the exact value of \(\mathrm{E}(D^2)\).

Find the exact values of \(\mathrm{E}(S)\) and \(\mathrm{E}(S^2)\).

Hence, by showing that \(D^2 = 4S^2 - 20S + 25\), verify the correctness of the value of \(\mathrm{E}(D^2)\) found in part (c).

2025 NYJC P2 Q7

2025 NYJC P2 Q7