Two unbiased dice are in the form of a regular tetrahedron and each has its faces numbered $1$, \(

Question

Two unbiased dice are in the form of a regular tetrahedron and each has its faces numbered $1$, $2$, $3$ and $4$. When each die is thrown on to a horizontal table, the number on the face in contact with the table is noted.

Suppose Tom and Harry play the following game: Tom throws the $2$ dice together and notes the total score, $X$. Harry tries to guess this total score and his guess is denoted by $Y$. If Harry guesses correctly, he wins $\$Y^{2}$, otherwise he loses $\$\frac{1}{2}Y$.

Obtain the probability distribution of $X$ and find $\mathrm{E}\left(\frac{1}{X}\right)$, correct to $3$ decimal places.
Harry guesses the total score to be $6$ in a particular game. Find the expected gain in this game.

Accepted Answer

(i) For $x=2,3,4,5,6,7,8$, $P(X=x)=\frac1{16}(1,2,3,4,3,2,1)$, respectively, and $\mathrm E(1/X)=0.227$. (ii) $\$4.3125$.

Discrete Random Variables (DRV) - Constructing and Using Discrete Probability Distributions

Discrete Random Variables (DRV) - Constructing and Using Discrete Probability Distributions

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