Question

A state is starting a lottery game. To enter this lottery, a player uses a machine that randomly selects six distinct numbers from among the first 30 positive integers. The lottery randomly selects six distinct numbers from the same 30 positive integers. A winning entry must match the same set of six numbers that the lottery selected. The entry fee is 1, each winning entry receives a prize amount of 500,000, and all other entries receive no prize. Calculate the probability that the state will lose money, given that 800,000 entries are purchased.

Answer 1

Consider a randomly selected player.

The player can choose six distinct numbers from the first 30 positive integers in ways. Here, we are considering the order of the selection.

The player will win only if the machine selects the same set of distinct numbers, which is possible only in 6! ways.

Hence, the probability of the win of that player is

[Alternatively, a set of six distinct numbers can be selected from 30 numbers in ways and hence the probability that the lottery will choose the same is inverse of that]

Let X be the number of winning entries.

The entries are made randomly and probability of winning is the same for all.

Hence,

The state has earned 800000 as entry fees, and the lottery prize is 500000.

Hence, if the number of winning entries exceeds 2, the state will loose money.

Hence, required probability =

Using the R command pbinom(q = 1, size = 800000, prob = (1 / choose(30, 6)), lower.tail = FALSE), we get the probability as 0.3898443.