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.

• 0.33
• 0.39
• 0.61
• 0.67
• 0.74



Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

The state will receive 800,000($1) = $800,000 in revenue, and will lose money if there are 2 or more winning tickets sold. The player’s entry can be viewed as fixed. The probability the lottery randomly selects those same six numbers is from a hypergeometric distribution and is

[[math]] \frac{\binom{6}{6} \binom{24}{0}}{\binom{30}{6}} = \frac{1(1)}{\frac{30!}{6!24!}} = \frac{6(5)(4)(3)(2)(1)}{30(29)(28)(27)(26)(25)} = \frac{1}{593775} [[/math]]

The number of winners has a binomial distribution with n = 800,000 and p = 1/593,775. The desired probability is

[[math]] \begin{align*} \operatorname{P}(\textrm{2 or more winners}) &= 1 − \operatorname{P}(\textrm{0 winners}) − \operatorname{P}(\textrm{1 winner}) \\ &= 1 - \binom{800000}{0} \binom{1}{593775}^0 \binom{593774}{593775}^{800000} - \binom{800000}{1}\binom{1}{593775}^{700000} \\ &=1 − 0.2599 − 0.3502 =0.39. \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.