The following is an extract from a textbook (17C 6Ciii Extended Response):
Brett rides his bike to work each day. He knows that the time it takes is normally distributed with a mean of 55 minutes and a standard deviation of 5 minutes.
$$\therefore X \sim \mathcal{N}(55,5^{2})$$
During a five-day working week, Brett makes the ride 10 times. Find the probability that, in a randomly chosen week, the ride takes less than 50 minutes more than three times during the week.
So $Y \sim \mathcal{B}(10,Pr(X<50))$. I thought that the probability of "less than 50 minutes more than three times during the week" would be $Pr(Y>3)$, where the bounds are $4$ to $10$. But the book uses the bounds of $3$ to $10$ ($Pr(Y\ge3)$).
Why is $3$ included when the question specifically asks for more than three times?
Answer from textbook:
The textbook has either made a typo on the expression or the answer. The answer in decimal form is $Pr(Y\ge3) = 0.2037$, which is not $Pr(Y>3)$
There's either a typo or a mistake in the book.
Indeed, the statement $\Pr(X>3) = 1-\Pr(X\leq 2)$ is not true.