Puzzle | Bias and Unbias

Robin and Williams are playing a game. Repeatedly, a fair coin is tossed. As soon as the HHT throws sequence appears, Robin wins. As soon as the HTH toss sequence appears, Williams wins. When one of them wins, the game is over. What are the possibilities of each player winning?

Bias and Unbias

Solution:

Possible Outcomes: 8

1. T T T
2. HHH
3. HTT
4. THT
5. TTH
6. THH
7. HTH
8. HHT

Consider, 

The probability that HHT will appear = 1/8
The probability that HTH will appear = 1/8
The probability that both will not appear = 6/8

Robin can win the game in first trial or in second trial or in third.

So probability = 1/8 + (6/8 ×1/8) + (6/8×6/8×1/8) + (6/8×6/8×6/8×1/8)……
= sum of this infinite GP whose common ratio is 6/8 and first term is 1/8.
Sum = first term/(1- common ratio)
= (1/8) / (1 – 6/8 )
= 1/2

Probability for William: 1 – 1/3 = 2/3

2nd Process: Robin wins as soon as the sequence of HHT occurs, which means only in HHT Robin will win only, it means only in one outcome he’ll win, thus, the Total no. of times the coin is tossed = 3 & winning outcome = 1

P(W) = No. of outcomes of winning / Total no. of outcomes
P(W) = 1/3 ,

So the winning probability of Robin is 1/3, Hence, William win probability = 1- 1/3 = 2/3