Question:
If the odds against the occurrence of an event be 4 : 7, find the probability of the occurrence of the event.
Solution:
We know that,
If odds in favor of the occurrence an event are a:b, then the probability of an event to
occur is $\frac{a}{a+b}$ similarly, if odds are not in the favor of the occurrence an event are a:b, then the probability of not occurrence of the event
is $\frac{\mathrm{a}}{\mathrm{a}+\mathrm{b}}$
We also know that,
Probability of occurring $=1$ - the probability of not occurring
$=1-\frac{a}{a+b}$
$=\frac{b}{a+b}$
Given a = 4 and b = 7
Probability of occurrence $=\frac{7}{4+7}$
$=\frac{7}{11}$
Conclusion: Probability that the event occurs is $\frac{7}{11}$