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 one player gets their $r$th win exactly at that game, meaning $(r-1)$ wins came in the first $(r-1)$ games.
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