What is the reason why we add 0.5 and subtract 0.5 in the normal probability to approximate the binomal probability?

Question

For example, why does $P(X \leq a) = P(Y \leq a + 0.5)$ and $P(X \geq a) = P(Y \geq a - 0.5)$ where $X$ is the binomial random variable and $Y$ is the normal random variable?

Answer 1

Binomial is discrete (ie- 0,1,2,3,4,5...), yet Normal is continuous (can take any decimal figure). Thus, we convert the discrete probability range to a continuous (and cut the missing space between the digits in half). So a discrete P(X>1) would be a continuous P(X>0.5) since we fill the gap between 0 and 1. Remember discrete ranges CAN have gaps, yet continuous CANNOT. Hope this helps, Chris