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Discrete Mathematics

Discrete Probability

Discrete Probability: Level 3 Challenges


Able and Brainy are stumped by a multiple choice problem with three options and each of them takes a wild guess at it. When the proctor is distracted, they look at what answer the other has put down. Each of them has a fifty-fifty chance of either keeping his answer or copying his friend's. They get two more opportunities to crosscheck before they turn the test in. On both occasions, they behave exactly the same way, with the same probabilities.

What is the probability that they both have the correct answer at the end?

If the probability is in the form \( \frac mn\), where \(m\) and \(n\) are coprime positive integers, submit your answer as \(m+n\).

A box contains 4 balls. The color of each of the balls is one of the three: White, Black or Red. However, you don't know how many balls of each color are there. It might even be the case that all the balls are of the same color. You then draw two balls randomly from the box. The probability that both the drawn balls are red can be written as \(\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. What is the value of \(p+q\)?

Select two distinct positive integers (without order) between 1 and 29 inclusive. If the probability that their sum is divisible by 3 can be represented in the form \(\dfrac{a}{b}\) where \(a\) and \(b\) are coprime positive integers, compute \(a+b.\)

Three 6-sided fair dice are rolled together. Let \(P(n) \) denotes the probability of obtaining a total sum of \(n\). What is the relationship between \(P(9) \) and \(P(10) \)?

Amy and Blake are each dealt two cards from a standard deck. Given no additional information, the probability that Blake is dealt a pair is \(\frac{3}{51} = \frac{1}{17}\) since his second card needs to match his first card.

Given that Amy is dealt a pair, what is the probability that Blake has a pair?


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