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 continues, 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.

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]

⇒ [tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{A}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{A}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{A}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{A}[/tex] [tex]P_{A}[/tex]  + ................

= [tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex] + ........

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

so P(A is overall winner) will be;

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

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

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

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

= [tex]\frac{5}{16}[/tex] × [tex]\frac{8}{7}[/tex]

= 5/14 or 0.35714

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

The probability that A is the overall winner is; 5/14

Now, if the player loses, then it means that two games in a row are won by the same player, and then the tournament will be over. Thus; [tex]P_{L}[/tex] = 0

       [tex]P_{WLX}[/tex] = ¹/₈

Furthermore from that above, he will have another chance to win twice in a row which gives a probability in form of a geometric series as;

¹/₄ + (¹/₄ × ¹/₈ ) + (¹/₄ × ¹/₈² ) +  (¹/₄ × ¹/₈³ ) + ........

Now, for that series, as n approaches infinity, the sum of the series would give us;

∑ = ¹/₄(1/(1 - ¹/₈ ))

⇒ ¹/₄(⁸/₇)

⇒ ²/₇

Since player C's first game would already be against another game's winner, then it means that P is also the chance player C will win the tournament but the probability that player A will win is same as probability that player B will win which is;

[1 - (²/₇)]/2

⇒ ⁵/₇ × ¹/₂

P(A) = P(B) = ⁵/₁₄

Read more on probability at; https://brainly.com/question/24855677

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