 Quantitative Finance

# Discrete Conditional Probabilities

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?

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