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Suppose a fair coin is tossed $3$ times. Let the random variable $X$ represent the number of heads on the first and second tosses and let $Y$ represent the number of heads on the second and third tosses. Write the joint probability distribution of $X$ and $Y$.

I solved this problem by making a list, exhaustion, and am wondering if there is an easier way to do this. I can't seem to get it using combinations.

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The simplest way is to do the table, for such a small case. Write out all $2^3 = 8$ possible outcomes for the 3 tosses of the coin, where $H = 1$ and $T = 0$: $$\{0,0,0\}, \quad \{0,0,1\}, \\ \{0,1,0\}, \quad \{0,1,1\}, \\ \{1,0,0\}, \quad \{1,0,1\}, \\ \{1,1,0\}, \quad \{1,1,1\}. \\$$ Then write out the table counting $X$ and $Y$: $$\begin{array}{|c|ccc|}\hline &0&1&2 \\ \hline 0 & \tfrac{1}{8} & \tfrac{1}{8} & 0 \\ 1 & \tfrac{1}{8} & \tfrac{1}{4} & \tfrac{1}{8} \\ 2 & 0 & \tfrac{1}{8} & \tfrac{1}{8} \\ \hline \end{array}$$

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