Discrete Mathematics
# Conditional Probability

You were invited to play a game. The game is played with a 6-sided fair dice.

You choose two different numbers from the dice, and then the dice is rolled.

If the result is one of the numbers you chose, you win a dollar, otherwise, you lose one dollar.

You knew it is an unfair game, but you wanted to try your luck, so you decided to play with two dollars. You ended up losing your two dollars, but the game owner decided to lend you as much money as you want to spend in the game. You decided to play until you recover your two dollars, and without owing anything to the owner.

Let \(P\) be the probability that you achieve your goal, find \(\left\lfloor {10^{10} e^P + 0.5} \right\rfloor \).

A student is **admired by his peers** if he has a rating that is higher than strictly more than half of his friends (out of these 600 students). Out of these 600 students, what is the most number of students that can be admired?

For similar problems, you can read my note on Construction.

There are four boxes. Each contains 2 balls. The first box has a red and a white ball in it. The remaining three boxes each have two white balls in them.

A ball is picked at random from box 1 and put in box 2.

Then a ball is picked at random from box 2 and put into box 3.

Then a ball is picked at random from box 3 and put into box 4.

Finally, a ball is picked from box 4.

The probability that the ball picked from box 2 is red, given that the final ball picked from box 4 is white can be written as \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

More probability questions

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Initially, a large urn contains 3 black marbles and 1 white marble. Every minute, a marble is chosen at random from the urn, and then returned to the urn, together with another marble of the same colour.

If the probability that, after exactly one hour, precisely three-quarters of the marbles in the urn are black can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

In front of you are two coins. They are identical in appearance, but one is fair (when flipped, it comes up Heads 50% of the time), while the other comes up Heads 75% of the time. Your task is to determine which one is the unfair coin.

You will only be permitted two flips total. After you choose your first coin and flip it, you can base your decision of which coin to flip second on your results of the first flip. After you perform your second flip, you will be asked to declare which coin you believe to be the fake.

If you always act so as to maximize the likelihood of your eventually correctly identifying the fake coin, what is the probability that you *will* correctly identify the fake coin?

If the desired probability is written as \(\frac{m}{n}\), where \(m\) and \(n\) are positive coprime integers, then enter \(m+n\) as your answer.

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