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This really has me stumped! The Washington state lottery has a new game called Zip Bingo. Every ticket costs $2 and consists of 2 regular Bingo cards with 35 call numbers. The prizes are as follows:

Regular bingo on card 1: $2

Regular bingo on card 2: $3

Regular bingo on both card 1 and card 2: $5

Match 4 corners on card 1: $10

Match 4 corners on card 2: $15

Match X pattern on card 1: $25

Match X pattern on card 2: $35

Match 4 corners on card 1 and X pattern on card 2: $45

Match Z pattern on card 1: $100

Match Z pattern on card 2: $200

Blackout on card 1: $500

Blackout on card 2: $20,000

Only one prize per ticket.

I tried to calculate the average return from this game, and in order to simplify things just considered the regular bingo and the four corners. It seems a little bit tricky because the events are not mutually exclusive.

The probability of a bingo with 35 numbers is 0.271983.

The probability of getting all four corners with 35 numbers is 0.043078695.

So:

(0.271983)*((1-0.043078695)^2)*2 = 0.50

(0.271983)*((1-0.043078695)^2)*3 = 0.75

(0.043078695)*(1-0.043078695)*10 = 0.41

(0.043078695)*15 = 0.65

0.50 + 0.75 + 0.41 + 0.65 = $2.31 but each ticket only costs $2!

So, where did I go wrong in my math?

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# Mathematical expectation of Zip Bingo

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