This is a binomial distribution problem with the following parameters:
- Number of trials: n = 32
- Probability of success: p = probability that a character is a number = 10/16 = 5/8
- Probability of failure: q = 1 - p = probability that a character is a letter = 6/16 = 3/8
We want to find the probability that at least 20 characters are numbers. We can use the complement rule and find the probability that fewer than 20 characters are numbers, and subtract that from 1:
P(at least 20 numbers) = 1 - P(fewer than 20 numbers)
Using the binomial probability formula, we have:
P(fewer than 20 numbers) = Σ[k=0 to 19] C(n,k) * p^k * q^(n-k)
where C(n,k) is the binomial coefficient "n choose k". This can be calculated using a calculator or software that has a binomial probability function. For example, in Python, we can use the scipy.stats.binom.cdf() function:
from scipy.stats import binom
n = 32
p = 5/8
q = 3/8
prob_fewer_than_20 = binom.cdf(19, n, p)
This gives prob_fewer_than_20 = 0.04903110366420758.
Therefore,
P(at least 20 numbers) = 1 - 0.04903110366420758 = 0.9509688963357924
Rounding to three decimal places, the answer is 0.951.