# Probability that a 5 occurs first

Discussion in 'Mathematics' started by Jimmy Sabater, Oct 8, 2018.

1. ### Jimmy SabaterGuest

Suppose we roll pair of dice until a sum of either 5 or 7 appears. What is the probability that a sum of 5 occurs first?
Try:

Let $A$ be the event that a sum of $5$ occurs on the ith roll and $B$ that the sum of 7 occurs on the ith roll. We are interested on the event $A | A^c \cup B^c$. We have

$$P(A | A^c \cup B^c) = \dfrac{ P(A \cap (A^c \cup B^c))}{P(A^c \cup B^c)} = \frac{P(A \cap B^c)}{1 - P(A \cap B)} = \frac{P(A \cap B^c)}{1-0} = P(A \cap B^c)$$

Is this approach correct so far?