Two standard six-sided dice are tossed. A diagram representing the sample space is shown below. Let&

Question

Two standard six-sided dice are tossed. A diagram representing the sample space is shown below. Let $X$ be the sum of the scores on the two dice. Second die First die 123456 1$(1,1)$$(1,2)$$(1,3)$$(1,4)$$(1,5)$$(1,6)$ 2$(2,1)$$(2,2)$$(2,3)$$(2,4)$$(2,5)$$(2,6)$ 3$(3,1)$$(3,2)$$(3,3)$$(3,4)$$(3,5)$$(3,6)$ 4$(4,1)$$(4,2)$$(4,3)$$(4,4)$$(4,5)$$(4,6)$ 5$(5,1)$$(5,2)$$(5,3)$$(5,4)$$(5,5)$$(5,6)$ 6$(6,1)$$(6,2)$$(6,3)$$(6,4)$$(6,5)$$(6,6)$ Find[6] $P(X = 6)$; $P(X > 6)$; $P(X = 7 | X > 5)$. Elena plays a game where she tosses two dice. If the sum is 6, she wins 3 points. If the sum is greater than 6, she wins 1 point. If the sum is less than 6, she loses k points. Find the value of k for which Elena’s expected number of points is zero.[7]

Accepted Answer

Zero expected score requires $3(\frac5{36})+1(\frac{21}{36})-k(\frac{10}{36})=0$, so $k=3.6$.

Second die First die	1	2	3	4	5	6
1	\((1,1)\)	\((1,2)\)	\((1,3)\)	\((1,4)\)	\((1,5)\)	\((1,6)\)
2	\((2,1)\)	\((2,2)\)	\((2,3)\)	\((2,4)\)	\((2,5)\)	\((2,6)\)
3	\((3,1)\)	\((3,2)\)	\((3,3)\)	\((3,4)\)	\((3,5)\)	\((3,6)\)
4	\((4,1)\)	\((4,2)\)	\((4,3)\)	\((4,4)\)	\((4,5)\)	\((4,6)\)
5	\((5,1)\)	\((5,2)\)	\((5,3)\)	\((5,4)\)	\((5,5)\)	\((5,6)\)
6	\((6,1)\)	\((6,2)\)	\((6,3)\)	\((6,4)\)	\((6,5)\)	\((6,6)\)

Use a two-dice sample space for conditional probability and construct a zero-expectation game.

Use a two-dice sample space for conditional probability and construct a zero-expectation game.

Solution:

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