Final answer:
To calculate the probabilities for different values of a binomial random variable X, use the binomial probability formula P(X = x) = C(n, x) * (p^x) * (q^(n-x)) with the given n, x, and p values, performing calculations and rounding to three decimal places.
Step-by-step explanation:
The question at hand deals with calculating probabilities using the binomial distribution. A binomial random variable X represents the number of successes in n independent trials, with each trial having two possible outcomes: success (with probability p) and failure (with probability q, where q=1-p).
To calculate the probability of x successes, we use the binomial probability formula:
P(X = x) = C(n, x) * (p^x) * (q^(n-x)),
where C(n, x) represents the combination of n items taken x at a time.
- For n = 4, x = 2, p = 0.55: P(2) = C(4, 2) * (0.55^2) * (0.45^2)
- For n = 5, x = 0, p = 0.55: P(0) = C(5, 0) * (0.55^0) * (0.45^5)
- For n = 7, x = 5, p = 0.55: P(5) = C(7, 5) * (0.55^5) * (0.45^2)
- For n = 7, x = 3, p = 0.15: P(3) = C(7, 3) * (0.15^3) * (0.85^4)
- For n = 1, x = 0, p = 0.7: P(0) = C(1, 0) * (0.7^0) * (0.3^1)
- For n = 9, x = 4, p = 0.2: P(4) = C(9, 4) * (0.2^4) * (0.8^5)
The calculations must be done using the combination formula and the relevant probability powers, and then the results are rounded to three decimal places.