Discrete Probability
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?
Submit your answer as a decimal.