Solutions - 5010-02

(a)

(1) Divide the sample space into two disjoint sets:

Let ${\mathrm{\Omega }}_1$ be the sample space where the die result is even, and ${\mathrm{\Omega }}_2$ where it is odd. Then $\mathrm{\Omega }={\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2$.

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(2) Describe ${\mathrm{\Omega }}_1$:

There are $3$ even numbers on a die, and $2$ possible results of each of $2$ coin flips, so $|{\mathrm{\Omega }}_1|=3\cdot 2^2=12$.

$${\mathrm{\Omega }}_1=\{(i,j,k):i\in \{2,4,6\},j,k\in \{H,T\}\}$$

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(3) Describe ${\mathrm{\Omega }}_2$:

There are $3$ odd numbers on a die and $2$ possible results of $1$ coin flip, so $|{\mathrm{\Omega }}_2|=3\cdot 2=6$.

$${\mathrm{\Omega }}_2=\{(i,j):i\in \{1,3,5\},j\in \{H,T\}\}$$

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(4) Combine to describe $\mathrm{\Omega }$:

Since ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are disjoint, $|\mathrm{\Omega }|=12+6=18$.

$$\mathrm{\Omega }={\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2=\{(i_1,j_1),(i_2,j_2,k):i_1\in \{1,3,5\},i_2\in \{2,4,6\},j_1,j_2,k\in \{H,T\}\}$$

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(b)

$$|𝒫(\mathrm{\Omega })|=2^{|\mathrm{\Omega }|}=2^{18}=262144$$