I'm studying statistics through Khan Academy and it had this question:
"Yuki and Zana are on a swimming team. They often compete against each other in the 100 meter freestyle race. Yuki's times in this race are normally distributed with a mean of 80 seconds and a standard deviation of 4.2 seconds. Zana's times are also normally distributed with a mean of 85 seconds and a standard deviation of 5.6 seconds. We can assume that their times are independent.
Suppose we choose a random 100-meter freestyle race and calculate the difference between their times.
Find the probability that Yuki's time is faster than Zana's."
Khan Academy's answer to the question is 0.7611 or approximately 0.76. I've spent hours working on this problem and I frustratingly keep on ending up with an answer of 0.2389. Can someone please help me on what I'm doing wrong?
I had the very same issue. As @mihaild mentions in the comment, your solution is the inverse of the true one. The real reason for this is in the text though.
What you (and I) calculated was the probability that Yuki's time is greater than Zana's. However, the question is about the probability that Yuki's time is faster, not greater. Because, of course, she is faster when her time is lower.