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Mathematical expectation of Zip Bingo

  1. Aug 5, 2005 #1
    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? :confused:
     
  2. jcsd
  3. Aug 6, 2005 #2

    EnumaElish

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    "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?
     
  4. Aug 6, 2005 #3
    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).
     
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