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Math Help - A tricky odds question URGENT

  1. #1
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    A tricky odds question URGENT

    In the king of fighters 2 tournament, Goku and Vegeta are the final 2 combatants and must face each other in the main event. The tournament winner is determined in a best of five matches series. If the odds of goku beating vegeta are 5 to 7, what is the probability Goku will win the tournament.?
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  2. #2
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    Hello, Faisal2007!

    There is no neat formula for this problem.
    We must list the possible outcomes . . .


    In the "King of Fighters 2" tournament, Goku and Vegeta are the final two combatants
    and must face each other in the main event.
    The tournament winner is determined in a best-of-five-matches series.
    If the odds of goku beating vegeta are 5 to 7,
    what is the probability Goku will win the tournament?

    We have: . P(G) = \tfrac{5}{12},\;\;P(V) = \tfrac{7}{12}


    Three-match series: Goku wins the first three matches.

    . . The probability is: . P(\text{3 matches}) \:=\:\left(\tfrac{5}{12}\right)^3


    Four-match series: Goku wins two of the first three and the fourth match.

    . . There are three ways: . VGGG,\:GVGG,\:GGVG

    . . The probability is: . P(\text{4 matches}) \:=\:3\left(\tfrac{5}{12}\right)^3\left(\tfrac{7}{  12}\right)


    Five-match series: Goku wins two of the first four and the fifth match.

    . . There are six ways: . VVGGG,\:VGVGG,\:VGGVG,\:GVVGG,\:GVGVG,\:GGVVG

    . . The probability is: . P(\text{5 matches}) \:=\:6\left(\tfrac{5}{12}\right)^3\left(\tfrac{7}{  12}\right)^2


    Therefore: . P(\text{Goku wins}) \:=\:\left(\tfrac{5}{12}\right)^3 + 6\left(\tfrac{5}{12}\right)^3\left(\tfrac{7}{12}\r  ight) + 6\left(\tfrac{5}{12}\right)^3\left(\tfrac{7}{12}\r  ight)^2

    . . . . . . . . . . . . . . . . . = \;\frac{86,\!250}{248,\!832} \;=\;\frac{14,\!375}{41,\!472} \;\approx\; 34.7\%

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