Question

The odds against A solving a certain problem are $4$ to $3$, and the odds in favour of B solving the same problem are $7$ to $5$: what is the chane that the problem will be solved if they both try?

Answer 1

Let the two events be $E_A$ and $E_B$

Odds against favour of one $A$ are $4:3$

$\Rightarrow$ odds in favour of $A$ are $3:4$

$\Rightarrow$ probability that a solves the problem $=P\left(E_A\right)=\frac{3}{7}$

$\Rightarrow$probaility that $A$ does not solve the problem $=P\left(\stackrel{\_}{E_A}\right)=1-\frac{3}{7}=\frac{4}{7}$

Odds in favour of $B$ are $7:5$

$\Rightarrow$ probability that $B$ solves the problem $=P\left(E_B\right)=\frac{7}{12}$

$\Rightarrow$ probability that $B$ does not solve the problem $=P\left(\stackrel{\_}{E_B}\right)=1-\frac{7}{12}=\frac{5}{12}$

Probability that problem is solved $=P\left(E\right)=1-$none of them solves the problem

$\Rightarrow P\left(E\right)=1-P\left(\stackrel{\_}{E_A}\right)P\left(\stackrel{\_}{E_B}\right)\Rightarrow P\left(E\right)=1-\frac{4}{7}\times \frac{5}{12}=\frac{64}{84}$