ipip207 asked in 科學及數學數學 · 1 decade ago

數學問題2: 概率的Expected value

請教一下各位:

原題為: What is the expected minimum number of coin tosses you would need to make in order to get HTT ( head, then tail, then another tail)?

在"摘公字"(coin toss)中, 要連續摘出公,字,字( 即H,T,T) 的最小期望值為多少?

謝謝解答。

3 Answers

Rating
  • 1 decade ago
    Favorite Answer

    For notation, call Ex the expected number of throws required to finish the game from position x. Thus E0 is the expected game length starting from scratch, Eh is the expected number of throws remaining after throwing an H, and Eht The expected number of throws after throwing an HT, and so on.

    -- Regarding the HTT game --

    At the beginning, we expect that E0 throws remain, and the value of E0 is what we need to find. After one throw either Eh or Et throws to remain, with a .5 chance of each possibility.

    [0] E0 = (1+Eh)/2 + (1+Et)/2

    Also [1] Et=E0

    E0 = (1+Eh)/2 + (1+E0)/2

    2E0=1+Eh+1+E0

    E0=2+Eh

    Now proceeding similarly, one throw after being in position H we will wind up in HH or HT. And for the HTT game, we can note that note that Ehh is the same as E0:

    [2] Ehh=Eh

    Eh = (1+Ehh)/2 + (1+Eht)/2

    E0-2 = (1+E0-2)/2 + (1+Eht)/2

    2E0-4=E0-1+1+Eht

    E0=4+Eht

    [3] E0 = 4 + Eht

    Now proceeding as before, after HT we'll expect Ehth or Ehtt:

    [4] Eht = (1+Ehth)/2 + (1+Ehtt)/2

    And since HTT is the end of the game, and HTH puts us right back at the beginning:

    [5] Ehtt = 0

    [6] Ehth = Eh = E0 - 2

    Hence combing [4], [5], [6]:

    [7] Eht = (1+E0-2)/2 + 1/2

    And substituting [7] into [4]:

    E0 = 4 + (E0-1)/2 + 1/2

    2E0 = 8 + E0 - 1 + 1

    E0 = 8

    Thus the HTT game requires, on average, 8 throws to complete.

    2008-05-14 23:57:31 補充:

    http://www2.bc.edu/~rosen/public/CoinFlips.pdf

    最general

  • 1 decade ago

    total possible output

    = 2x2x2

    = 8

    HTT is 1 of the output.

    therefore, possibility of getting HTT is 1/8

    thus, the expected value to get HTT is 8.

    That is, we EXPECT to get at least one HTT if we toss the coin 8 times.

  • 1 decade ago

    最小期望值為1/8,8分之1,12.5%

Still have questions? Get your answers by asking now.