Question

Let a fair die be rolled 2 times. Let’s assume that the 2 rolls are independent. Let X and Y be the outcomes of the first and second rolls, respectively.
a. What is the probability distribution of X+Y? That is, create a table that contains each unique possible value of X+Y (each value only listed once) and each possibility’s corresponding probability.
b. What is the probability that X+Y is greater or equal to 10?

Answer 1

Answer on Question #43271 – Math - Statistics and Probability

Let a fair die be rolled 2 times. Let's assume that the 2 rolls are independent. Let $X$ and $Y$ be the outcomes of the first and second rolls, respectively.

a. What is the probability distribution of $X + Y$ ? That is, create a table that contains each unique possible value of $X + Y$ (each value only listed once) and each possibility's corresponding probability.

b. What is the probability that $X + Y$ is greater or equal to 10?

Solution

The distribution of one outcome is

We calculate joint distribution $\mathrm{P}(\mathrm{X} = \mathrm{k},\mathrm{Y} = \mathrm{l}),\mathrm{k},\mathrm{l} = 1,\dots,6.$ If $\mathrm{X}$ and $\mathrm{Y}$ are independent random variables then $\mathrm{P}(\mathrm{X} = \mathrm{k},\mathrm{Y} = \mathrm{l}) = \mathrm{P}(\mathrm{X} = \mathrm{k})\mathrm{P}(\mathrm{Y} = \mathrm{l})$ . The joint distribution of $\mathrm{X}$ and $\mathrm{Y}$ is

We calculate possibilities of outcome $X + Y$ . For value 2 we have one way when $X = 1$ and $X = 1$ only. For 3 we have two ways $X = 1$ and $Y = 2$ or $X = 2$ and $Y = 1$ .

We calculate probability multiply possibility to probability with possibilities of outcome of $X + Y$ .

b) $P(X + Y \geq 10) = \frac{1}{12} + \frac{1}{18} + \frac{1}{36} = \frac{1}{6}$

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