There are 6 white and 4 black balls in the urn. One ball is removed from the urn and then returned to the urn

There are 6 white and 4 black balls in the urn. One ball is removed from the urn and then returned to the urn. Find the probability that the white ball will appear 10 times out of 40.

1. Let:

A – the event that the white ball is removed;
B – the event that the black ball is removed;
X (n, k) – the event that the white ball appears in k cases out of n trials.
2. When returning the ball back to the urn, we get n independent tests of event A with the same probability p, so we can use the Moivre-Laplace formula:

P (n, k) = r * φ (x), where
r = 1 / √ (npq);
x = (k – np) r;
φ (x) = e ^ (- x ^ 2/2) / √ (2π) is the Gaussian function.
3. We have:

n = 40;
k = 10;
p = P (A) = 6 / (6 + 4) = 6/10 = 0.6;
q = P (B) = 1 – p = 0.4.
Let’s calculate the probability of the event X (n, k):

r = 1 / √ (npq) = 1 / √ (40 * 0.6 * 0.4) ≈ 0.3227;
x = (k – np) r = (10 – 40 * 0.6) r = (10 – 24) r = -14r ≈ -4.5185;
P (40, 10) = r * φ (x) ≈ r * φ (-4.5185) ≈ 0.3227 * 1.4705 * 10 ^ (- 5) ≈ 4.7463 * 10 (-6).
Answer: 4.7463 * 10 (-6).