Answer:
The probability that there is at least one new case of the disease is approximately 0.19121
Step-by-step explanation:
Given that the probability of contracting a disease is p = 1/27124.
In an experiment n = 5756, we want to find the probability that there will be at least one new case of the disease. That is P(X ≥ 1).
If x is approximated Binomial(n, p)
Then
P(X = x) = (nCx)(p^x)q^(n - x)
Where q = 1 - p
Here, q = 1 - (1/27124) = 27123/27124
And nCx, read as "n combination x"
is given as n!/(n - x)! x!
Also note that
P(X ≥ a) = 1 - P(X < a)
So, it is sufficient to find P(X < 1) for this problem.
P(X < 1) = P(X = 0)
= (5756C0)(1/27124)^0(27123/27124)^(5756 - 0)
= 1 × 1 × 0.80879
≈ 0.80879
Now,
P(X ≥ 1) = 1 - 0.80879 = 0.19121
