<p>A biased four-sided die with faces numbered ${1}$, ${2}$, ${3}$ and ${4}$ is used in a game. The random variable ${X}$ represents the number shown on the face which lands on the table.</p>
<p>The probability of landing on a ${4}$ is ${0.05}$ and the probability of landing on a ${3}$ is ${0.11}$. There is equal probability of landing on either a ${1}$ or a ${2}$.</p>
<p>In a game, players are to throw the die twice and the two independent observations ${{X}_{1}}$ and ${{X}_{2}}$ produce the score ${Y}$ defined as</p>
<p style="text-align: center">$Y=\left\{ \begin{matrix} 4\,\,\,\,\,\,\,\,\mathrm{if}\,\,\,\,\,\,\,\,{{X}_{1}}=4, \\ {{X}_{1}}+{{X}_{2}}\,\,\,\,\,\,\,\,\mathrm{if}\,\,\,\,\,\,\,\,{{X}_{1}}\ne 4. \\\end{matrix} \right.$</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P}\left( Y=y \right)$ for all possible values of $y$.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{E}\left( Y \right)$ and $\mathrm{Var}\left( Y \right)$.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>Players pay ${\$5}$ for a game. If the score $Y$ is ${5}$ or ${6}$, the player wins ${\$20}$. If the score $Y$ is $7$, the player wins ${\$50}$, otherwise the player wins no money. In a typical day, a player plays $40$ games.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the expected loss of the player in a typical day.</span></div><p style="text-align: right;">[2]</p></li></ol>

A biased four-sided die with faces numbered ${1}$, ${2}$, ${3}$ and ${4}$ is used in a game. The random variable ${X}$ represents the number shown on the face which lands on the table.

The probability of landing on a ${4}$ is ${0.05}$ and the probability of landing on a ${3}$ is ${0.11}$. There is equal probability of landing on either a ${1}$ or a ${2}$.

In a game, players are to throw the die twice and the two independent observations ${{X}_{1}}$ and ${{X}_{2}}$ produce the score ${Y}$ defined as

$Y=\left\{ \begin{matrix} 4\,\,\,\,\,\,\,\,\mathrm{if}\,\,\,\,\,\,\,\,{{X}_{1}}=4, \\ {{X}_{1}}+{{X}_{2}}\,\,\,\,\,\,\,\,\mathrm{if}\,\,\,\,\,\,\,\,{{X}_{1}}\ne 4. \\\end{matrix} \right.$

Find $\mathrm{P}\left( Y=y \right)$ for all possible values of $y$.

Find $\mathrm{E}\left( Y \right)$ and $\mathrm{Var}\left( Y \right)$.

Players pay ${\$5}$ for a game. If the score $Y$ is ${5}$ or ${6}$, the player wins ${\$20}$. If the score $Y$ is $7$, the player wins ${\$50}$, otherwise the player wins no money. In a typical day, a player plays $40$ games.

2023 NYJC P2 Q7

2023 NYJC P2 Q7

Video Solution: