in a essay competition the odds against competitors a,b,c,d is 2:1,3:1,4:1 and 5:1 respectively .find the probability that one of them wins the competition
Hello, bluffmaster.roy.007!
In a competition the odds against competitors $\displaystyle A,B,C,D$ winning are: .$\displaystyle 2\!:\!1,\;3\!:\!1,\;4\!:\!1,\;5\!:\!1$, resp.
Find the probability that one of them wins the competition.
We have: .$\displaystyle \begin{array}{|c|c|c|c|} \hline
\text{Entrant} & \text{odds against} & P(\text{win}) & P(\text{lose}) \\ \hline \hline \\[-4mm]
A & 2:1 & \frac{1}{3} & \frac{2}{3} \\ [2mm]
B & 3:1 & \frac{1}{4} & \frac{3}{4} \\ [2mm]
C & 4:1 & \frac{1}{5} & \frac{4}{5} \\ [2mm]
D & 5:1 & \frac{1}{6} & \frac{5}{6} \\ [2mm] \hline \end{array}$
I will assume that there is exactly one winner of the competition.
. . $\displaystyle \begin{array}{ccccc}P(\text{A wins}) &=& \frac{1}{3}\cdot\frac{3}{4}\cdot\frac{4}{5}\cdot\f rac{5}{6} &=& \frac{1}{6} \\ \\
P(\text{B wins}) &=& \frac{1}{4}\cdot\frac{2}{3}\cdot\frac{4}{5}\cdot\f rac{5}{6} &=& \frac{2}{9} \\ \\
P(\text{C wins}) &=& \frac{1}{5}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot\f rac{5}{6} &=& \frac{1}{12} \end{array}$
. . $\displaystyle \begin{array}{ccccc}P(\text{D wins}) &=& \frac{1}{6}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot\f rac{4}{5} &=& \frac{1}{15} \end{array}$
Therefore: .$\displaystyle P(\text{A or B or C or D wins}) \;=\; \frac{1}{6} + \frac{2}{9} + \frac{1}{12} + \frac{1}{15} \;=\;\frac{97}{180}$