In one lottery, a player wins the jackpot by matching all five distinct numbers drawn in any order from the white balls(1 through 41) and matching the number on the gold ball (1 through 34). If one ticket is purchased, what is the probability of winning the jackpot?
There are a total of 41 white balls, and the player needs to match 5 of them in any order. The number of ways to choose 5 distinct numbers from a set of 41 is given by the combination formula: C(41, 5) = 41! / (5! * (41 - 5)!). This simplifies to: 41! / (5! * 36!). Similarly, there are 34 possible choices for the gold ball.
The probability of winning the jackpot is the number of ways to win divided by the total number of possible outcomes, which is the product of the number of ways to choose 5 white balls and the number of choices for the gold ball:
P(jackpot) = [C(41, 5) * 34] / [41! / (5! * 36!)]
= [41! * 34!] / [(5! * 36!) * 5! * 34]
= 41! / (5! * 36! * 5)
= (41 * 40 * 39 * 38 * 37) / (5 * 4 * 3 * 2 * 1 * 36 * 35)
≈ 0.000000862
Therefore, the probability of winning the jackpot is approximately 0.000000862, or 1 in 1,158,764.