Probability 問題!!唔該晒

P(the former happens first)
= P(HH)
= 1/2 * 1/2 = 1/4

P(the latter happens first)
= P(TH) + [P(HTH) + P(TTH)] + [P(HTTH) + P(TTTH)] + ...
= 1/2 * 1/2 + [(1/2)^3 + (1/2)^3] + [(1/2)^4 + (1/2)^4] + ...
= 1/4 + 2 * [(1/2)^3 + (1/2)^4 + ...]
= 1/4 + 2 * (1/2)^3 / (1 - 1/2) [Sum of G.P. to infinity when 0 < r < 1]
= 1/4 + 2 * (1/2)^2
= 1/4 + 2 * 1/4
= 3/4

So, B has a better chance of winning.

It is a very interesting question!!! At first glance, it seems that both A and B have same chance winning, but it is not true. Actually, B has better chance winning this.

Since someone else have put a mathematically solution to this, I will try to answer it in other way… hope you find this interesting ^_^

Firstly, we have to make 3 very important yet interesting conclusions…

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#1: The game cannot end up with a draw. If A doesn’t wins, B MUST win eventually
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It seems to be an obvious conclusion, but it is very important.

The game can only draw when the outcome is HTTT… or TTTT… (T goes on forever in both cases).

Both case have probability 0 of happening provided that they would have play long enough

So if A doesn’t win, B must win. Bear this in mind.


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#2: If A wins, he have to win before the first T comes out
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A cannot win after a T comes out.

For A to win, he must get 2 consecutive H. But after a T come out, A will not have the chance to wait for his 2 consecutive H. Because when a first H come out, B will have won already! (T--->H)

So whenever a T comes out, A will not win


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#3: If a T comes out, B must win
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That is concluded from #1 and #2. After a T comes out, A cannot win. But the game cannot end up with a draw, so B must win



OK! After drawing these important conclusions, the solutions can be arrived in many ways.

Here is one of those methods…

The outcome of first 2 toss can be…

(i) HH …… (Winner: A, by rules)
(ii) HT …… (Winner: B, by #3)
(iii) TH …… (Winner: B, by rules)
(iv) TT …… (Winner: B, by #3)

P(A wins) = 0.25
P(B wins) = 0.75

B has better chance winning.

SAME opportunity
a. 1/2*1/2=1/4
b.1/2*1/2=1/4