Final Answer:
(a) P(x ≤ 3) = 0.8257
(b) P(x > 1) = 0.9650
(c) P(x < 1) = 0.0064
(d) P(x ≥ 2) = 0.7179
Step-by-step explanation:
For each case, we need to use the binomial probability formula:
P(x) = nCx * p^x * (1-p)^(n-x)
where:
n is the number of trials
p is the probability of success in each trial
x is the number of successes we are interested in
nCx is the binomial coefficient, which can be calculated using a calculator or statistical software
Calculations:
(a) P(x ≤ 3) = 4C3 * 0.3^3 * 0.7 + 4C2 * 0.3^2 * 0.7^2 + 4C1 * 0.3^1 * 0.7^3 + 4C0 * 0.3^0 * 0.7^4 ≈ 0.8257
(b) P(x > 1) = 1 - P(x ≤ 1) = 1 - (6C0 * 0.1^0 * 0.9^6 + 6C1 * 0.1^1 * 0.9^5) ≈ 0.9650
(c) P(x < 1) = P(x = 0) = 9C0 * 0.8^0 * 0.2^9 ≈ 0.0064
(d) P(x ≥ 2) = 1 - P(x < 2) = 1 - (8C0 * 0.1^0 * 0.9^8 + 8C1 * 0.1^1 * 0.9^7) ≈ 0.7179
These calculations provide the probabilities for each case based on the given parameters.