Conditional Probability on type of coin

  1. CAF123

    CAF123 2,434
    Gold Member

    1. The problem statement, all variables and given/known data
    Suppose we have 10 coins such that if the ith coin is flipped, heads will appear with probability i/10, i = 1,2...10. When one of the coins is randomly selected and flipped, what is the conditional probability that it was the coin?


    2. Relevant equations
    Bayes's Formula


    3. The attempt at a solution
    First of all, I can't make any sense out of how choosing a different coin will give a different probability of showing a head?
    In solving the problem, I used Bayes's relation; [tex] P(\text{5th coin} | \text{heads}) = \frac{P(\text{heads | 5th coin})P(\text{5th coin})}{P(\text{heads| 5th coin})P(\text{5th coin}) + P(\text{heads | not 5th coin})P(\text{not 5th coin})} [/tex]

    where [tex] P(\text{heads |5th coin}) = \frac{5}{10}, P(\text{5th coin}) = \frac{1}{10}, P(\text{heads | not 5th coin}) = \frac{P(\text{heads and not 5th coin})}{P(\text{not 5th coin})} = \frac{\frac{9}{10}\frac{1}{2}}{\frac{9}{10}} = \frac{1}{2}, P(\text{not 5th coin}) = 1-\frac{1}{10}. [/tex] Putting this together gives the wrong answer. Any ideas?
     
  2. jcsd
  3. Ray Vickson

    Ray Vickson 6,313
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    Homework Helper

    There is something missing from your problem statement: you say "When one of the coins is randomly selected and flipped, what is the conditional probability that it was the coin?" Did you mean that one of the coins was flipped and came up heads? Did you mean 'what is the conditional probability it was coin i?" I will assume the answer is YES to both of these questions.

    P{coin i|H} = P{coin i & H}/P{H}. What is the numerator equal to? How do you find the denominator?

    RGV
     
  4. CAF123

    CAF123 2,434
    Gold Member

    Oh sorry, how careless of me. It should read 'When one of the coins is randomly selected and flipped, it shows heads. What is the conditional probability that it was the 5th coin'
     
  5. Ray Vickson

    Ray Vickson 6,313
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    OK, so my previous post is correct if we put i = 5. Now can you answer the questions I asked there? Take it one step at a time. Apply Bayes' rules, etc.

    RGV
     
  6. CAF123

    CAF123 2,434
    Gold Member

    I got that P(coin 5|head) = P(head|coin5)P(coin5)/ƩP(head|coin i)P(coin i). (Where the sum is from i=1 to i=5). Is this correct so far?
     
  7. Ray Vickson

    Ray Vickson 6,313
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    We have 10 coins, not 5.

    RGV
     
  8. CAF123

    CAF123 2,434
    Gold Member

    I think P(head/coin 5) = 5/10 and P(coin i ) = 1/10 since all equally likely to be picked. I then said [itex] \sum_{i=1}^{5} P(head|coin\, i)P(coin\, i) = (1/10)(1/10) + (2/10)(1/10) + (3/10)(1/10) + (4/10)(1/10) + (5/10)(1/10) + (6/10)(1/10) + (7/10)(1/10) + (8/10)(1/10) + (9/10)(1/10) + (1/10) [/itex] I now get the right result - thanks for your help!

    EDIT : let i go form 1 to 10.
     
    Last edited: Oct 18, 2012
  9. Ray Vickson

    Ray Vickson 6,313
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    What's so special about the number '5'? Suppose, instead, I asked you for the conditional probability of coin 2, or coin 7 or coin 10. How would you express the conditional probabilities in those cases? Remember, I first asked you about the conditional probability of coin i, where I did not specify i.

    RGV
     
  10. Ray Vickson

    Ray Vickson 6,313
    Science Advisor
    Homework Helper

    OK now, but of course you need to say ##\sum_{i=1}^{10},## which is actually what you calculated.

    RGV
     
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