(i)

P(both G and H wins) = P(G wins) and P(H wins)

Note: and means mutliplication

P(G wins) = $\frac{1}{5}$

P(H wins) = $\frac{3}{8}$

P(both G and H wins) = $\frac{1}{5}$ x $\frac{3}{8}$

P(both G and H wins) = $\frac{\mathrm{1\; x\; 3}}{\mathrm{5\; x\; 8}}$ = $\frac{3}{40}$

(ii)

P(Only one contract wins) = P(G wins and H loses) or P(G loses and H wins)

Note: and means multiplication and or means addition

P(Not) = 1 - P(Is)

P(G loses) = 1 - P(G wins)

P(G loses) = 1 - $\frac{1}{5}$

Every number is being divided by 1

1 = $\frac{1}{1}$

P(G loses) = $\frac{1}{1}$ - $\frac{1}{5}$

P(G loses) = $\frac{\mathrm{5\; x\; 1\; -\; 1\; x\; 1}}{5}$

P(G loses) = $\frac{\mathrm{5\; -\; 1}}{5}$

P(G loses) = $\frac{4}{5}$

P(H loses) = 1 - P(H wins)

P(H loses) = 1 - $\frac{3}{8}$

P(H loses) = $\frac{1}{1}$ - $\frac{3}{8}$

P(H loses) = $\frac{\mathrm{8\; x\; 1\; -\; 1\; x\; 3}}{8}$

P(H loses) = $\frac{\mathrm{8\; -\; 3}}{8}$

P(H loses) = $\frac{5}{8}$

P(Only one contract wins) = P(G wins and H loses) or P(G loses and H wins)

P(G wins) = $\frac{1}{5}$

P(H loses) = $\frac{5}{8}$

P(H wins) = $\frac{3}{8}$

P(G loses) = $\frac{4}{5}$

P(Only one contract wins) = $\frac{1}{5}$ x $\frac{5}{8}$ + $\frac{4}{5}$ x $\frac{3}{8}$

P(Only one contract wins) = $\frac{\mathrm{1\; x\; 5}}{\mathrm{5\; x\; 8}}$ + $\frac{\mathrm{4\; x\; 3}}{\mathrm{5\; x\; 8}}$

P(Only one contract wins) = $\frac{5}{40}$ + $\frac{12}{40}$

P(Only one contract wins) = $\frac{\mathrm{5\; +\; 12}}{40}$

P(Only one contract wins) = $\frac{17}{40}$