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?
You throw two dice at the same time. The first die shows \( 5, \) and we don't know what the other die shows. Then what is the probability that the sum of outcomes of the two dice is \( 7? \)
If two exclusive events \(A\) and \(B\) satisfy
\[ P\left(A \cap B^C\right) = \frac15, \quad P\left(A^C \cap B\right) = \frac14, \]
what is the value of \( P(A \cup B) \)?
There are two identical boxes \(A\) and \(B.\) In box \(A,\) there are two red balls and one white ball. In box \(B,\) there are three red balls and two white balls. If you randomly chose one box and drew a white ball out of it, what is the probability that you chose box \(A?\)
When you roll two fair dice and the product \(p\) of the values is four, what is the conditional probability distribution for the value \(X\) of the first dice? Represent it in the form of \[ (P(X=1|p=4),P(X=2|p=4),\dots,P(X=6|p=4)) .\]
Justin is taking out balls one by one from a box which contains \( 6 \) red balls and \( 5 \) blue balls. After he takes out \( 3 \) balls from the box, he checks their colors: \( 2 \) red and \( 1 \) blue. What is the probability that the first ball he took out was red?