A biased four-sided die, $\mathrm{A}$, is rolled. Let $X$ be the score obtained when die \(

Question

A biased four-sided die, $\mathrm{A}$, is rolled. Let $X$ be the score obtained when die $\mathrm{A}$ is rolled. The probability distribution for $X$ is given in the following table. $x$$1$$2$$3$$4$ $\mathrm{P}(X = x)$$p$$p$$p$$\frac{1}{2}p$ Find the value of $p$. [2] Hence, find the value of $\mathrm{E}(X)$. [2] A second biased four-sided die, $\mathrm{B}$, is rolled. Let $Y$ be the score obtained when die $\mathrm{B}$ is rolled. The probability distribution for $Y$ is given in the following table. $y$$1$$2$$3$$4$ $\mathrm{P}(Y = y)$$q$$q$$q$$r$ [3] State the range of possible values of $r$. Hence, find the range of possible values of $q$. Hence, find the range of possible values for $\mathrm{E}(Y)$. [3] Agnes and Barbara play a game using these dice. Agnes rolls die $\mathrm{A}$ once and Barbara rolls die $\mathrm{B}$ once. The probability that Agnes’ score is less than Barbara’s score is $\frac{1}{2}$. Find the value of $\mathrm{E}(Y)$. [6]

Accepted Answer

(a) $p = \frac{2}{7}$ (b) $\mathrm{E}(X) = \frac{16}{7}$ (c)(i) $0 \leq r \leq 1$ (c)(ii) $0 \leq q \leq \frac{1}{3}$ (d) $2 \leq \mathrm{E}(Y)\leq 4$ (e) $\mathrm{E}(Y) = \frac{11}{4}$

2021 May TZ1 P1 Q9

2021 May TZ1 P1 Q9