in a huddle race the odds that A will win are 2 to 3 and the odds that B will win are 1 to 2. Find the probability 
(i)  A or B wins the race, assuming there is no tie.
(ii) ratio of odds that A or B coins the race.

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Odds} \mathrm{of} \mathrm{A} \mathrm{win} =2:3 \\ \mathrm{Odds}=\frac{\mathrm{Probability} \mathrm{of} \mathrm{A} \mathrm{win}}{\mathrm{Probability} \mathrm{of} \mathrm{A} \mathrm{loose}}=\frac{{\displaystyle \frac{2}{\mathrm{x}}}}{{\displaystyle \frac{3}{\mathrm{x}}}} \\ \mathrm{since}, \\ \frac{2}{\mathrm{x}}+\frac{3}{\mathrm{x}}=1 \\ \mathrm{x}=5 \\ \mathrm{Probability} \mathrm{of} \mathrm{A} \mathrm{win}=\frac{2}{5} \\ \mathrm{Odds} \mathrm{of} \mathrm{B} \mathrm{win}=1:2 \\ \mathrm{similarly}, \\ \mathrm{Probability} \mathrm{of} \mathrm{B} \mathrm{win}=\frac{1}{3} \\ \mathrm{P}\left(\mathrm{Awin}\right)=\frac{2}{5} \\ \mathrm{P}\left(\mathrm{Bwin}\right)=\frac{1}{3} \\ \mathrm{P}\left(\mathrm{Awin} \mathrm{or} \mathrm{Bwin}\right)=\mathrm{P}\left(\mathrm{Awin}\right)\times \mathrm{P}\left(\mathrm{Blooses}\right)+\mathrm{P}\left(\mathrm{Alooses}\right)\times \mathrm{P}\left(\mathrm{Bwin}\right)+\mathrm{P}\left(\mathrm{Awin}\right)\times \mathrm{P}\left(\mathrm{Bwin}\right)=\frac{2}{5}\times \frac{2}{3}+\frac{3}{5}\times \frac{1}{3}+\frac{2}{5}\times \frac{1}{3}=\frac{6}{15}+\frac{1}{5}=\frac{9}{15} \\ \mathrm{odds} \mathrm{of} \mathrm{A} \mathrm{win} \mathrm{or} \mathrm{B} \mathrm{win}=\frac{{\displaystyle \frac{9}{15}}}{1-{\displaystyle \frac{9}{15}}}=\frac{9}{6}=\frac{3}{2}=3:2 \end{gathered}$$

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