Suppose you log on to an online chess website and are randomly assigned a match against an opponent. Suppose at any time on the site,

Question

Suppose you log on to an online chess website and are randomly assigned a match against an opponent.
Suppose at any time on the site, 30% of opponents are rated ovice”, 55% are rated mid”, and 15% are
rated high”. In a chess match, a player can either win, lose, or draw. Suppose that you win 75% and
draw 10% of matches against novice rated opponents, you win 50% and draw 20% of matches against
mid rated opponents, and you win 10% and draw 25% of matches against high rated opponents.
(a) For a randomly assigned match, what is the probability that you play someone ranked high and
lose?
(b) For a randomly assigned match, what is the probability that you win?
(c) You play a match and win. What is the probability that you played a high rated opponent?

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Kim Cúc 5 years 2021-07-17T03:51:21+00:00 1 Answers 80 views 0

Answers ( )

  1. danthu
    0
    2021-07-17T03:52:41+00:00
    Reply

    Answer:

    a) 0.1125 = 11.25% probability that you play someone ranked high and lose.

    b) 0.5435 = 54.35% probability that you win

    c) 0.0690 = 6.90% probability that you played a high rated opponent

    Step-by-step explanation:

    Conditional Probability

    We use the conditional probability formula to solve this question. It is

    P(B|A) = \frac{P(A \cap B)}{P(A)}

    In which

    P(B|A) is the probability of event B happening, given that A happened.

    P(A \cap B) is the probability of both A and B happening.

    P(A) is the probability of A happening.

    (a) For a randomly assigned match, what is the probability that you play someone ranked high and lose?

    15% probability that you play someone that is ranked high.

    If you play someone ranked high, 100 – 25 = 75% probability you lose

    0.15*0.75 = 0.1125

    0.1125 = 11.25% probability that you play someone ranked high and lose.

    (b) For a randomly assigned match, what is the probability that you win?

    75% of 30%(novice oponent)

    50% of 55%(mid oponent)

    25% of 15%(high opponent). So

    p = 0.77*0.3 + 0.5*0.55 + 0.25*0.15 = 0.5435

    0.5435 = 54.35% probability that you win

    (c) You play a match and win. What is the probability that you played a high rated opponent?

    Here, we use the conditional probability formula.

    Event A: Winning

    Event B: Playing a high opponent.

    0.5435 = 54.35% probability that you win

    This means that P(A) = 0.5435

    Intersection of events A and B:

    25% of 15%(high opponent). So

    P(A \cap B) = 0.25*0.15 = 0.0375

    Question:

    P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.0375}{0.5435} = 0.0690

    0.0690 = 6.90% probability that you played a high rated opponent

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