Independent solution

How to solve this Expected Value question

Setup

Setup

Count the possible five-number sets, before considering the distinguished position.

M=(305)=142,506M=\binom{30}{5}=142{,}506

Model

Model

A fixed entry receives the base award when its number set matches. The extra award additionally requires its distinguished number to match.

Pr⁡(base match)=1M\Pr(\text{base match})=\frac1M
Pr⁡(extra match)=15M\Pr(\text{extra match})=\frac1{5M}

Compute

Compute

Use linearity for the two award components and then for all entries.

E[Wentry]=50,000M+250,0005ME[W_{\mathrm{entry}}]=\frac{50{,}000}{M}+\frac{250{,}000}{5M}
E[Wmonth]=100,000(100,000142,506)E[W_{\mathrm{month}}]=100{,}000\left(\frac{100{,}000}{142{,}506}\right)
E[Wmonth]=70,172.48397…E[W_{\mathrm{month}}]=70{,}172.48397\ldots

Answer

Answer

The expected monthly payout rounds to 70,172.

70,172(A)\boxed{70{,}172\quad\text{(A)}}