Calculate probabilities from binomial or normal distribution

Question

I have the following experiment:

Two balls are let fall from a certain high (could be the same or different)

Then for each of them, I get a random number from 0 to 1 with a uniform distribution and multiply the original height. The result is the height that will bounce back.

The whole process is repeated 10 times and note down the number of ball A ended up higher.

An example:

ball_A_Original_Height = 10
ball_B_Original_Height = 10
wins = 0
repeat 10 times:
    new_ball_A_Height = get_random_number(0,1) * ball_A_Original_Height // 0.81 * 10 = 8.1

    new_ball_B_Height = get_random_number(0,1) * ball_B_Original_Height // 0.57 * 10 = 5.7

    if new_ball_A_Height >= new_ball_B_Height then wins += 1

Then I run a simulation with 5.000.000 times the above test and create a dataset with the number of wins from ball A.

If I use an online probability calculator for a Binomial Distribution like this one stattrek, with probability of success 50%, I get that P(X<=5) = 62.30%

But if I use a probability calculator for a Normal Distribution stattrek-normal with the mean and standard deviation of the dataset I created, the same probability P(X<=5) is far less.

Can someone explain it to me?

Answer 1

The binomial score has to deal with the case of exactly equal to 5, the normal approximation assumes a continuous variable, so the probability of equaling exactly 5 is 0.