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 and . Bag contains white balls and black balls while bag contains white balls and 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 being selected to draw the balls is . Find the value of .
On planet Brilliantia, there are two types of creatures: mathematicians and non-mathematicians.
Mathematicians tell the truth of the time and lie only of the time, while non-mathematicians tell the truth of the time and lie of the time.
It is also known that there is a chance a creature from Brilliantia is a mathematician and a 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 , for some coprime positive integers , find the value of .
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?