# Thread: A tricky odds question URGENT

1. ## 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.?

2. 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: .$\displaystyle P(G) = \tfrac{5}{12},\;\;P(V) = \tfrac{7}{12}$

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

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

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

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

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

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

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

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

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

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