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Conditional Probability

Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago? See more

Level 4

         

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 \(A\) and \(B\). Bag \(A\) contains \(n\) white balls and \(2\) black balls while bag \(B\) contains \(2\) white balls and \(n\) 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 \(A\) being selected to draw the balls is \(\frac{6}{7}\). Find the value of \(n\).

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

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

It is also known that there is a \(\frac{2}{3}\) chance a creature from Brilliantia is a mathematician and a \(\frac{1}{3}\) 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 \(\frac{a}{b}\), for some coprime positive integers \(a, b\), find the value of \(b - a\).

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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