A biased coin is tossed repeatedly. The probability of landing a head is p. The game ends the first time when two consecutive heads (HH) or two consecutive tails (TT) are observed. A player wins if (HH) is observed and loses if (TT) is observed. Find the probability that he wins. (For example if the outcome is THTHTT, he loses. On the other hand, if the outcome is s THTHHH,he wins)
In this question of probability ,I am a bit confused with the language of the question. Can someone help me out. What does the problem Want and how to frame that probability?I mean how to proceed off with the probkem
Can anyone kindly help me out
Calculate the probability that the game ends with HH. If the game begins with H, the possible outcomes are: HH, HTHH, HTHTHH, HTHTHTHH, ... But if the game begins with T, the possible outcomes are: THH, THTHH, THTHTHH, ... So, the probability of two heads from the first outcome is $$P(HH)(1+P(HT)+P(HTHT)+P(HTHTHT)+\cdots )$$ While the probability of winning after starting with T is $$P(THH)(1+P(TH)+P(THTH)+\cdots)$$
This yields:
$$(P(HH)+P(THH))\cdot \sum_{k=0}^\infty (P(HT))^k = \dfrac{P(HH)+P(THH)}{1-P(HT)}$$