The probability of an event occurring in each of the independent trials is 0.7.

The probability of an event occurring in each of the independent trials is 0.7. Find the number of trials n for which the most probable number of occurrences of events is 20.

1. Let:

A – some event;
p = P (A) = 0.7;
q = 1 – p = 0.3;
n is the number of trials of event A;
k is the number of occurrences of event A;
kmax = 20.
2. Let’s use the formula of Moivre-Laplace:

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. For a specific value of n, the most probable number of occurrences of events kmax will be provided:

x = 0;
(kmax – np) r = 0;
kmax – np = 0;
kmax = np, hence:
n = kmax / p = 20 / 0.7 = 200/7 ≈ 28.6;
We take the closest integer to this value:

n = 29.
Answer: 29.