Question 1200427: Suppose it is known that the probability of recovery for a certain disease is 0.4. If 35 people are stricken with the disease, what is the probability that:
(a) 25 or more will recover?
(b) Fewer than five will recover?
(Use the normal approximation.)
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! **1. Define the parameters:**
* **p:** Probability of recovery = 0.4
* **n:** Number of people stricken with the disease = 35
**2. Calculate the mean and standard deviation:**
* **Mean (μ):** μ = n * p = 35 * 0.4 = 14
* **Standard deviation (σ):** σ = √(n * p * (1 - p)) = √(35 * 0.4 * 0.6) ≈ 2.898
**3. (a) Probability of 25 or more recovering:**
* **Continuity correction:** Since we're using the normal approximation to the binomial distribution, we need to apply a continuity correction. For "25 or more," we'll use 24.5 as the lower bound.
* **Calculate the z-score:**
z = (X - μ) / σ = (24.5 - 14) / 2.898 ≈ 3.62
* **Find the probability:**
Using a standard normal distribution table or calculator, we find the area to the right of z = 3.62.
P(X ≥ 25) ≈ 0.00015
**4. (b) Probability of fewer than 5 recovering:**
* **Continuity correction:** For "fewer than 5," we'll use 4.5 as the upper bound.
* **Calculate the z-score:**
z = (X - μ) / σ = (4.5 - 14) / 2.898 ≈ -3.28
* **Find the probability:**
Using a standard normal distribution table or calculator, we find the area to the left of z = -3.28.
P(X < 5) ≈ 0.0005
**Therefore:**
* (a) The probability that 25 or more people will recover is approximately **0.00015 (or 0.015%)**.
* (b) The probability that fewer than 5 people will recover is approximately **0.0005 (or 0.05%)**.
**Note:** These probabilities are based on the normal approximation to the binomial distribution. For a more precise calculation, you could use the binomial probability formula directly.
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