Final answer:
The probability of more than 3 positive tests in the next 200 tests is approximately 97.85%.
Step-by-step explanation:
To find the probability of more than 3 positive tests in the next 200 tests, we can use the binomial probability formula.
The formula is:
P(X > k) = 1 - P(X ≤ k)
where P(X > k) is the probability of getting more than k successes, P(X ≤ k) is the probability of getting at most k successes, and n is the total number of tests.
In this case, the probability of getting more than 3 positive tests in the next 200 tests can be calculated as:
P(X > 3) = 1 - P(X ≤ 3)
To calculate P(X ≤ 3), we can use the binomial cumulative distribution function. The formula is:
P(X ≤ k) = ΣC(n, k) * pᵏ * (1-p)⁽ⁿ⁻ᵏ⁾
where C(n, k) is the number of combinations of n items taken k at a time, p is the probability of success in a single test, and n is the total number of tests.
In this case, p = 44/5110 = 0.0086 and n = 200.
Using the formula,
- P(X ≤ 3) = C(200, 0) * (0.0086)⁰ * (1-0.0086)⁽²⁰⁰⁻⁰⁾ + C(200, 1) * (0.0086)¹* (1-0.0086)⁽²⁰⁰⁻¹⁾ + C(200, 2) * (0.0086)^2 * (1-0.0086)⁽²⁰⁰⁻²⁾ + C(200, 3) * (0.0086)³ * (1-0.0086)⁽²⁰⁰⁻³⁾
- P(X ≤ 3) ≈ 0.0215.
Therefore, P(X > 3) = 1 - 0.0215 = 0.9785, or 97.85%.