$P_{1},P_{2},P_{3},\ldots.,P_{8}$ are 8 players participating in a tournament. If $i

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$P_{1},P_{2},P_{3},\ldots.,P_{8}$ are 8 players participating in a tournament. If $i < j$, then $P_{i}$ will win the match against $P_{j}$. Players are paired up randomly for first round and winners of this round again paired up for the second and so on. Then the probability that

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Sol. [Key Concept: Random-pairing knockout of $P_{1}..P_{8}$: compute reach-final / win probabilities by pools.](A) pool-A pool-B $$P_{4}$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$P = \frac{\ ^{4}C_{3}}{\ ^{7}C_{3}} = \frac{4}{35}$$(B) $P = 0$ (obvious)(C) $P = 1$ (obvious)(D) Pool-A Pool-B Pool -C Pool-D $$P_{6}$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$	imes$$ $$P = \frac{\ ^{2}C_{1}}{\ ^{7}C_{1}} = \frac{2}{7}$$ $$P = \frac{\ ^{2}C_{1}}{\ ^{7}C_{1}} = \frac{2}{7}$$

pool-A	pool-B
$$P_{4}$$	$$\times$$
$$\times$$	$$\times$$
$$\times$$	$$\times$$
$$\times$$	$$\times$$

Pool-A	Pool-B	Pool -C	Pool-D
$$P_{6}$$ $$\times$$	$$\times$$	$$\times$$	$$\times$$
	$$\times$$	$$\times$$	$$\times$$

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