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OlympiadBench / 2333 / Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

answer type

Expression

is multiple answer

false

language

English

question

Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each other. The other two players also play each other. The winners of the two matches then play to decide the tournament champion. Alain, Bianca and Chen are equally matched (that is, when a match is played between any two of them, the probability that each player wins is

\frac{1}{2}

). When Dave plays each of Alain, Bianca and Chen, the probability that Dave wins is

p

, for some real number

p

. Determine the probability that Bianca wins the tournament, expressing your answer in the form

\frac{a p^{2}+b p+c}{d}

where

a, b, c

, and

d

are integers.

Plain-text mathematical notation (without MathML)

Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each other. The other two players also play each other. The winners of the two matches then play to decide the tournament champion. Alain, Bianca and Chen are equally matched (that is, when a match is played between any two of them, the probability that each player wins is (1)/(2) ). When Dave plays each of Alain, Bianca and Chen, the probability that Dave wins is p, for some real number p. Determine the probability that Bianca wins the tournament, expressing your answer in the form (ap²+bp+c)/(d) where a,b,c, and d are integers.

Original LaTeX notation

Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each other. The other two players also play each other. The winners of the two matches then play to decide the tournament champion. Alain, Bianca and Chen are equally matched (that is, when a match is played between any two of them, the probability that each player wins is $\frac{1}{2}$ ). When Dave plays each of Alain, Bianca and Chen, the probability that Dave wins is $p$, for some real number $p$. Determine the probability that Bianca wins the tournament, expressing your answer in the form $\frac{a p^{2}+b p+c}{d}$ where $a, b, c$, and $d$ are integers.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2333
task_id: 212625a1-31f3-520b-b671-bb0a93a403ea
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2333
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import