Probability
Probability
Allen Star Batch
Grade 12
Question:
Players $P_1, P_2, P_3, \ldots, P_n$ of equal skill, play a game consecutively in pairs as $P_1P_2, P_2P_3, P_3P_4, \ldots, P_nP_1, \ldots$ and any player who wins two consecutive games (i.e $k$ and $(k+1)$th game) wins the match. If the chance that the match is won at the $r$th game is $k$ then:
$k = \frac{3}{8}$, if $r = 5$
$k = \frac{3}{32}$, if $r = 5$
$k = \frac{3}{32}$, if $r = 6$
$k = \frac{5}{64}$, if $r = 6$
Step-by-Step Solution
Key Concept: The match ends at game r when a player wins consecutive games for the first time. For this to happen at exactly game r, the first (r-1) games must not contain any two consecutive wins, and games (r-1) and r must both be won by the same player. This requires counting valid sequences where no two consecutive games have the same winner until game r.
The match ends when one player wins $r$ games. For the match to end at the $r$th game, there must be exactly $(r-1)$ favorable outcomes of the form where wins and losses alternate. Each case has probability $\frac{1}{2} \times \frac{1}{2} \times ... \times \frac{1}{2}$ (appearing $r$ times), giving $\frac{1}{2^r}$. With $(r-1)$ such favorable cases total, the required probability is $\frac{r-1}{2^r}$.
Correct Answer: 1,4