Discrete Probability
If two six-sided dice are rolled, the probability that they both show the same number can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?
A fair 6-sided die is rolled twice. The probability that the second roll is strictly less than the first roll can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are positive, coprime integers. What is the value of \(a+b\)?
Details and assumptions
The roll of a dice refers to the value on the top face of the dice.
Two players each flip a fair coin. The probability that they get the same result can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?
Lily has two bags containing balls. The first bag contains \(10\) balls, with each ball labelled by a distinct number from \(1\) through \(10\). The second bag contains one ball labelled \(1\), two balls labelled \(2\), etc, up to ten balls labelled with \(10\). Suppose Lily draws one ball from each bag uniformly at random and let \(\frac{a}{b}\) be the probability that the two balls have the same value, where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?
Some friends are sitting together playing a game that involves rolling dice. On one turn, a player rolls six 6-sided dice and gets one of each number showing. Another player sees this and asks "what are the chances of that?" If the probability of rolling six different numbers on six 6-sided dice can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a + b\)?