Final answer:
To find the probabilities, we can use the binomial distribution formula. The probability of the antibiotic being effective in exactly 3 individuals is approximately 0.309. The probability of it being effective in all five individuals is approximately 0.168. The probability of it being effective in none of the five individuals is also approximately 0.168. The mean number of patients in whom the antibiotic is effective is 3.5, and the variance is 0.315.
Step-by-step explanation:
To calculate the probabilities, we can use the binomial distribution formula. Let's define some variables:
- n = number of individuals (5 in this case)
- p = probability of success (70% or 0.7)
- x = number of individuals for which the antibiotic is effective (3 for part a, 5 for part b)
a. To find the probability that the antibiotic is effective in exactly 3 individuals, we can use the formula: P(X = x) = (nCx) * (p^x) * ((1-p)^(n-x)). Plugging in the values, we get: P(X = 3) = (5C3) * (0.7^3) * ((1-0.7)^(5-3)). Simplifying, we get P(X = 3) = 10 * 0.343 * 0.09 = 0.3087, or approximately 0.309.
b. To find the probability that the antibiotic is effective in all five individuals, we can use a similar formula: P(X = x) = (nCx) * (p^x) * ((1-p)^(n-x)). Plugging in the values, we get: P(X = 5) = (5C5) * (0.7^5) * ((1-0.7)^(5-5)). Simplifying, we get P(X = 5) = 1 * 0.16807 * 1 = 0.16807, or approximately 0.168.
c. To find the probability that the antibiotic is effective in none of the five individuals, we can again use the formula: P(X = x) = (nCx) * (p^x) * ((1-p)^(n-x)). Plugging in the values, we get: P(X = 0) = (5C0) * (0.7^0) * ((1-0.7)^(5-0)). Simplifying, we get P(X = 0) = 1 * 1 * 0.16807 = 0.16807, or approximately 0.168.
d. The mean (expected) number of patients in whom the antibiotic is effective can be found using the formula: E(X) = n * p. Plugging in the values, we get: E(X) = 5 * 0.7 = 3.5. Therefore, the expected number of patients in whom the antibiotic is effective is 3.5.
The variance in the number of successes can be calculated using the formula: Var(X) = n * p * (1-p). Plugging in the values, we get: Var(X) = 5 * 0.7 * (1-0.7) = 1.05 * 0.3 = 0.315. Therefore, the variance in the number of successes is 0.315.