Discrete Mathematics

# Probability - Rule of Product

There are $$6$$ white beads and $$5$$ black beads in your pocket. You randomly pull the beads one by one out of your pocket and place them on a table. Let $$\frac{a}{b}$$ be the probability that the third bead drawn is the first white bead, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b?$$

Two independent, fair 6 sided dice are rolled. The first die face shows a $$6$$. The second die rolls under the table, so you can not see its face. The probability that the dice under the table is $$6$$ given that the first dice is $$6$$ can be written as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are positive coprime integers. What is the value of $$a + b$$?

Liam has 4 black shirts and 6 white shirts in his closet. He also has 5 black ties and 3 white ties in his drawer. If Liam reaches into the closet and takes out a shirt at random, then reaches into the drawer and takes out a tie at random, then the probability that the shirt and the tie are the same color is $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?

A bag contains $$3$$ red marbles, $$4$$ blue marbles, $$6$$ yellow marbles, and $$6$$ green marbles. Four marbles are randomly drawn from the bag, without replacement. The probability of drawing a red, then blue, then green, and finally another red marble is $$\frac{a}{b}$$, where $$a$$ and $$b$$ are positive coprime integers. What is the value of $$a+b$$?

The probability that it will rain on any given day in August in Los Angeles is 9%. What is the probability that it rains on both August 13, 2021 and August 13, 2022?