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Pani-rosa [81]
3 years ago
5

Problem 3) A, B, and C are evenly matched tennis players. Initially A and B play a set and the winner then plays C. This continu

es, with the winner always playing the waiting player, until one of the players has won two sets in a row. That player is then declared the overall winner. Find the probability that A is the overall winner.
Mathematics
1 answer:
motikmotik3 years ago
3 0

Answer:

the probability that A is the overall winner is 5/14 or 0.35714

Step-by-step explanation:

Given the data in the question;

probability that A is the overall winner P[A is the overall winner]

⇒ P_{A}P_{A} + P_{A}P_{C}P_{B}P_{A}P_{A} + P_{A}P_{C}P_{B}P_{A}P_{C}P_{B}P_{A} P_{A}  + ................

= P_{B}P_{C}P_{A}P_{A} + P_{B}P_{C}P_{A}P_{B}P_{C}P_{A}P_{A} + P_{B}P_{C}P_{A}P_{B}P_{C}P_{B}P_{C}P_{A} + ........

P(A) = \frac{1}{2} = P_{B} = P_{C}             ∴{ they are evenly matched player}

so P(A is overall winner) will be;

= (\frac{1}{2})² + (\frac{1}{2})⁵ + (\frac{1}{2})⁸ + ..........+ (\frac{1}{2})⁴ + (\frac{1}{2})⁷ + (\frac{1}{2})¹⁰ + .............

= (\frac{1}{2})² [1 + (\frac{1}{2})³ + (\frac{1}{2})⁶ + .........] + (\frac{1}{2})⁴[1 + (\frac{1}{2})³ + (\frac{1}{2})⁶ + .........]

= [ (\frac{1}{2})² + (\frac{1}{2})⁴ ] [ 1 / 1 - (\frac{1}{2})³ ]

= [ (\frac{1}{4}) + (\frac{1}{16}) ] [ \frac{8}{7} ]

= \frac{5}{16} × \frac{8}{7}

= 5/14 or 0.35714

Therefore, the probability that A is the overall winner is 5/14 or 0.35714

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