## 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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6 months 2021-07-17T03:51:21+00:00 1 Answers 42 views 0

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

In which

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

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

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

Intersection of events A and B:

25% of 15%(high opponent). So

Question:

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