Solutions - 5190-01

(a)

Fill the cells using the respective column sum or row sum:

$P[B=2,G=0]$: We have $P[G=0]=0.375$, so

$$P[B=2,G=0]=0.375-0.15-0.1-0.0375\gg \gg 0.0875$$

$P[B=2,G=1]$: We have $P[G=1]=0.3875$, so

$$P[B=2,G=1]=0.3875-0.1-0.175\gg \gg 0.1125$$

$P[B=0,G=2]$: We have $P[B=0]=0.375$, so

$$P[B=0,G=2]=0.375-0.15-0.1-0.0375\gg \gg 0.0875$$

$P[B=1,G=2]$: We have $P[B=1]=0.3875$, so

$$P[B=1,G=2]=0.3875-0.1-0.175\gg \gg 0.1125$$

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(b)

(1) Add up the probabilities in which either $B=1$ or $G=1$:

$$\begin{array}{c}P[B=1\text{ or }G=1]\\ \\ \gg \gg 0.1+0.1+0.175+0.1125+0.1125\gg \gg 0.6\end{array}$$

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(2) Alternatively, use inclusion-exclusion with the marginal sums:

$$\begin{array}{c}P[B=1]+P[G=1]-P[B=1\text{ and }G=1]\\ \\ \gg \gg 0.3875+0.3875-0.175\gg \gg 0.6\end{array}$$