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.
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?