# Mathematical expectation of Zip Bingo

1. Aug 5, 2005

### O Great One

Hello everyone,
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?

2. Aug 6, 2005

### EnumaElish

"Regular bingo" means a line, right? What if a line overlaps with 4 corners? I guess what I am asking is how you calculate your probabilities. Also, are you assuming independent draws? What if the draws are correlated?

3. Aug 6, 2005

### O Great One

1. Yes. A 'regular bingo' means a line.
2. Yes. A line could occur at the same time as 4 corners. That's why it is necessary to multiply the probability of getting a bingo by the probability of not getting four corners.
3. Yes, the draws are independent from ticket to ticket. There is one group of 35 numbers and 2 bingo cards on each ticket. The player uses the same group of 35 numbers for both cards.

I got the probability of getting a bingo with 35 numbers from the wizardofodds website. It required the use of a computer program. The probability of getting four corners is (35C4/75C4).