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Math Help - Calculating the odds

  1. #1
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    Calculating the odds

    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
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  2. #2
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    Hello, bluffmaster.roy.007!

    In a competition the odds against competitors A,B,C,D winning are: . 2\!:\!1,\;3\!:\!1,\;4\!:\!1,\;5\!:\!1, resp.

    Find the probability that one of them wins the competition.

    We have: . \begin{array}{|c|c|c|c|} \hline<br />
\text{Entrant} & \text{odds against} & P(\text{win}) & P(\text{lose}) \\ \hline \hline \\[-4mm]<br />
A & 2:1 & \frac{1}{3} & \frac{2}{3} \\ [2mm]<br />
B & 3:1 & \frac{1}{4} & \frac{3}{4} \\ [2mm]<br />
C & 4:1 & \frac{1}{5} & \frac{4}{5} \\ [2mm]<br />
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.


    . . \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} \\ \\<br />
P(\text{B wins}) &=& \frac{1}{4}\cdot\frac{2}{3}\cdot\frac{4}{5}\cdot\f  rac{5}{6} &=& \frac{2}{9} \\ \\<br />
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}

    . . \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: . 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}

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