 Probability

# 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?