Question

The odds against a certain event are 5 to 2 and the odds in favour of an other event independent of the former are 6 to 5. The probability that none of the events will happen is

• A. $\frac{52}{77}$
• B. $\frac{12}{77}$
• C. $\frac{25}{77}$
• D. $\frac{5}{77}$

Answer 1

The correct option is C $\frac{25}{77}$

Let's say odds against a certain event are 5 to 2 is case A: $P\left(A\right)=\frac{2}{7}$
And another case is favour of an other event are 6 to 5 is case B: $P\left(B\right)=\frac{6}{11}$
Probability that none of the events will happen is$=1-P\left(A\right)-P\left(B\right)+P(A\cap B)$
$=1-P\left(A\right)-P\left(B\right)+P\left(A\right)P\left(B\right)$
$=1-\frac{2}{7}-\frac{6}{11}+(\frac{2}{7}\times \frac{6}{11})$
$=\frac{25}{77}$