The odds against an event are 5 : 3 and the odds in favour of another independent event are 7 : 5. Find the probability that at least one of the two events will occur.

Answer: Option C

Let probability of the first event taking place be A and probability of the second event taking place be B.
Then
$$\begin{array}{rl}& P\left(A\right)=\frac{3}{5+3}\\ & =\frac{3}{8}\\ & P\left(B\right)=\frac{7}{7+5}\\ & =\frac{7}{12}\end{array}$$
The required event can be defined as that A takes place and B does not take place (A or B takes place and A does not take place or A takes place and B takes place.)
$$=\left[P\left(A\right)\left\{1-P\left(B\right)\right\}\right]+$$     $$\left[P\left(B\right)\left\{1-P\left(A\right)\right\}\right]+$$     $$\left[P\left(A\right)P\left(B\right)\right]$$
$$=\left[\frac{3}{8}\times \frac{5}{12}\right]+$$   $$\left[\frac{5}{8}\times \frac{7}{12}\right]+$$   $$\left[\frac{3}{8}\times \frac{7}{12}\right]$$
$$\begin{array}{rl}& =\frac{15+35+21}{96}\\ & =\frac{71}{96}\end{array}$$