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Probability Help

  1. Feb 5, 2006 #1
    There are 5 independent fair coin flips made,
    What is the probabiltiy that at least two heads among the first 3 flips and at least two tails among the last three flips?

    Except writing down all the possibilities, is there any other way to figure this out?
     
  2. jcsd
  3. Feb 6, 2006 #2

    HallsofIvy

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    First think of it as two separate problems. What is the probability of "at least 2 heads in 3 flips"? What is the probability of "at least 2 tails in 3 flips"?

    One thing to be careful about: the middle (third) flip is common to both!

    I would do it this way:
    a. Assume the middle flip is heads. What is the probability of getting "at least 1 head in the first 2 flips" (the middle head then gives you 2). Since the middle flip is heads you now need to answer "what is the probability of getting at least 2 tails in 2 flips". Since they must both happen, and are independent, multiply the probabilities.

    b. Assume the middle flip is tails. Now, to get 2 heads in the first 3 flips, you must get 2 heads in the first 2 flips. What is that probabilty? Since the middle flip is tails you also must find the probability of getting 1 tail in the last two flips. Again, multiply those. (Because of the symmetry, don't be surprised if you get the same answer as in a.)

    Since either a or b must happen, and they are mutually exclusive, the probability you want is the sum of those two.
     
  4. Feb 6, 2006 #3
    thanks a lot!
    this is a much better method!!
    I have never thought of setting the third flip to be a set outcome..
    Thanks!
     
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