Sample space for one disc from each of two bags, the distribution and expectation of the total, and the expected gain from a game

12 marks

Bag $A$ holds three discs numbered $1$, $2$ and $4$. Bag $B$ holds three discs numbered $5$, $6$ and $6$, as shown below.

One disc is taken at random from each bag. Each of the nine pairs of discs is equally likely. The table shows some of the outcomes as (number from bag $A$, number from bag $B$).

	$5$	$6$
$1$	$(1,5)$
$2$		$(2,6)$
$4$

Complete the table to show all nine outcomes.[2]

Let $T$ be the total of the two numbers taken.

List all the possible values of $T$.[2]
Find the probability of each value of $T$.[2]
Find $\mathrm{E}(T)$.[3]

In a game, a player gains $\$20$ when $T$ is odd and loses $\$12$ when $T$ is even. The game is played $27$ times. Find the expected total gain over the $27$ games.[3]

Solution:

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Answer:(a) $(1,5),(1,6),(1,6),(2,5),(2,6),(2,6),(4,5),(4,6),(4,6)$ (b) $6,\ 7,\ 8,\ 9,\ 10$ (c) $\mathrm{P}(6)=\dfrac{1}{9},\ \mathrm{P}(7)=\dfrac{1}{3},\ \mathrm{P}(8)=\dfrac{2}{9},\ \mathrm{P}(9)=\dfrac{1}{9},\ \mathrm{P}(10)=\dfrac{2}{9}$ (d) $\mathrm{E}(T)=8$ (e) a gain of 60 dollars

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	\(5\)	\(6\)
\(1\)	\((1,5)\)
\(2\)		\((2,6)\)
\(4\)