Each round Mike and Dean toss coin each. Mike tosses a not fair coin in which the probability to get heads is 0.6. Dean tosses a not fair coin in which the probability to get heads is 0.1. they toss the coins till they get the same results at the same time. What is the probability that there will be at most 5 rounds?

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Ramiro Sosa · Accepted Answer

Step 1The game ends when Mike and Dean toss different results.Mike: Heads   =  0.6  +  T  a  i  l  s  =  0.4Dean: Heads   =  0.1  +  T  a  i  l  s  =  0.9The probability of round being:Both Heads   =  0.60  ⋅  0.40  =  0.24Both Tails   =  0.10  ⋅  0.90  =  0.09Both Same   =  0.24  +  0.09  =  0.33Both Different   =  1  −  0.33  =  0.67Step 2What is the probability that of getting to round X?  1  =  100  % (first round is always played)  2  =  0.33 (both same)  3  =      0.33    2    =  0.1089 (both same, twice)  4  =      0.33    3    =  0.0359 (both same, 3x)  5  =      0.33    4    =  0.0118 (both same, 4x)  6  =      0.33    5    =  0.0039  (both same, 5x)What is the probability that there will be at most 5 rounds?This is simply 1 minus the probability of getting to round 6  (  1  −      0.33    5    )  =  0.99608

Winston Todd · Accepted Answer

Step 1The probability till Mike and Dean get the same results at most 5 rounds:The final results can be...      H    H       o  r         T    T  The probability that Mike and Dean get the same results in a round   =      0.6    ×      0.1    +      (    1    −    0.6    )    ×      (    1    −    0.1    )    =  0.42.Step 2The probability that Mike and Dean do not get the same results in a round   =      1    −    0.42    =  0.58Let X be the number of rounds until Mike and Dean get the same results.   X  ∼  G  e  o  (  0.42  ).  P  (  X  ≤  5  )  =  P  (  X  =  1  )  +  P  (  X  =  2  )  +  P  (  X  =  3  )  +  P  (  X  =  4  )  +  P  (  X  =  5  )  P  (  X  ≤  5  )  =  0.42  +  0.58  ×  0.42  +  (  0.58      )    2    ×  0.42  +  (  0.58      )    3    ×  0.42  +  (  0.58      )    4    ×  0.42  ≈  0.9344     (  c  o  r  r  .  t  o     4     d  .  p  .  )

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