Discrete Mathematics

Conditional Probability

Conditional Probability: Level 4 Challenges


One green ball, one blue ball, and two red balls are placed in a bowl. I draw two balls simultaneously from the bowl and announce that at least one of them is red.

What is the chance that the other ball I have drawn out is also red?

Box I contains 1 red ball and 2 white balls. Box II contains 2 red balls and 1 white ball. One ball is drawn randomly from box I and transferred to box II.

Then, a ball is drawn randomly from box II and it is red. What is the probability that the transferred ball was white?

There are two bags AA and BB. Bag AA contains nn white balls and 22 black balls while bag BB contains 22 white balls and nn black balls . One of the two bags is selected at random and two balls are drawn from it without replacement . We know that if both the balls drawn are white balls, the probability of bag AA being selected to draw the balls is 67\frac{6}{7}. Find the value of nn.

On planet Brilliantia, there are two types of creatures: mathematicians and non-mathematicians.

Mathematicians tell the truth 67\frac{6}{7} of the time and lie only 17\frac{1}{7} of the time, while non-mathematicians tell the truth 15\frac{1}{5} of the time and lie 45\frac{4}{5} of the time.

It is also known that there is a 23\frac{2}{3} chance a creature from Brilliantia is a mathematician and a 13\frac{1}{3} chance that it is a non-mathematician, but there is no way of differentiating from these two types.

You are visiting Brilliantia on a research trip. During your stay, you come across a creature who states that it has found a one line proof for Fermat's Last Theorem. Immediately after that, a second creature shows up and states that the first creature's statement was a true one.

If the probability that the first creature's statement was actually true is ab\frac{a}{b}, for some coprime positive integers a,ba, b, find the value of bab - a.

Calvin flips a fair coin before he sees a run of an odd number of heads, followed by a tail. What is the expected number of times Calvin must flip the coin?


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