Probability - Probabilities of Successive Events
Probability - Probabilities of Successive Events
8 marks
Kai and Lina play a sequence of rounds. Kai wins the first round with probability \(\dfrac{3}{5}\). For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is \(\dfrac{2}{3}\). The match ends as soon as a player wins two consecutive rounds.
- The first four possible winning sequences for Kai are \(KK\), \(LKK\), \(KLKK\) and \(LKLKK\), where \(K\) denotes a round won by Kai and \(L\) denotes a round won by Lina. Find the probability of each sequence.[2]
- Hence, using the sum to infinity of a geometric progression, find the probability that Kai wins the match.[4]
- Given that Lina wins the first round, find the probability that Kai wins the match.[2]
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Answer:(i) \(P(KK)=\dfrac{2}{5},\ P(LKK)=\dfrac{4}{45},\ P(KLKK)=\dfrac{2}{45},\ P(LKLKK)=\dfrac{4}{405}\) (ii) \(\dfrac{11}{20}\) (iii) \(\dfrac{1}{4}\)
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