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Posted by on Wednesday, December 26, 2007 at 2:36pm.

"A batch of 350 raffle tickets contains three winning tickets. You buy five tickets. What is the probability that you have (a) no winning tickets, (b) all of the winning tickets, (c) at least one winning ticket, and (d) at least one nonwinning ticket?"

I would like the answers to the questions above, but I'd more like to know which sample is n and which is p.

  • Statistics - , Wednesday, December 26, 2007 at 3:40pm

    The probability of success, p, is 3/350 = .00857 so (1-p) = .99142
    Your number of trials is five, n=5
    The probability of k successes in n trials is [binomial coeff n,k) p^k (1-p)^(n-k)

    row five of Pascal's triangle (my easy way to get binomial coefficients rather than doing n!/[k!(n-k)!] ) is

    1 5 10 10 5 1

    so for example the probability of ZERO successes in your five trials is:
    (1) (.00857)^0 (.99142)^5
    =(1)(1)(.958)
    =.958
    Try the rest after checking my arithmetic carefully.

  • Statistics - , Wednesday, December 26, 2007 at 4:05pm

    Ok. That helped me out a lot. I have another one, if your willing to give direction.

    One in five people in the U.S. own individual stocks. In a random sample of 12 people, what is the probability that the number owning individual stocks is (a) exactly four, (b) at least four, and (c) more than four.

    So, in this case, my p is .20, correct? and my n is 12?

  • Statistics - , Wednesday, December 26, 2007 at 4:12pm

    However, I am not sure the binomial distribution applies here. Your probability is not constant as you draw winning tickets. The trials are not independent because if you get one winning ticket, your chances of getting a second go down. There are only two left in the pot if you draw one winner.

    Therefore question a, NO winning tickets. Initially you have 350-3 =347losers, each time you take a loser out, you have one less:
    (347/350)(346/349)(345/348)(344/347)(343/346)
    = .958
    That works out the same because we did not really change the nature of the population by drawing losers, However if we pull a winner out, things will change
    Let's try for all(3) winners in our five trials.
    Your chances for the first one are
    5(3/350)
    Your chances for a second one are
    4(2/349)
    Your chances for a third and final one are
    3(1/348)
    Multiply those together to find the chance of getting all three:
    =8.47*10^-6 or about 8 out of a million
    Now try the rest.

  • Statistics - , Wednesday, December 26, 2007 at 4:17pm

    Yes, p = 0.2 and n = 12 for the stock one. In this case the trials are indeed independent so you can use binomial distribution safely.
    I am wondering if the chapter you are working on is about binomial distributions if the first problem is really intended to have non-independent trials. Strange

  • Statistics - , Wednesday, December 26, 2007 at 5:03pm

    Well the question in the first problem is earlier in the book than in the second. We did learn about the process that you used early on though.

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