Question

Suppose that you have a bag of marbles, some of which are red, some of which are blue, and some of which are green. If you take marbles out one at a time, look at them, and then put them back before drawing the next one, we can model this as a Binomial Distribution. Suppose you draw 4 marbles out of the bag, what is the probability of drawing 2 red marbles if the proportion of red marbles is 0.22? Round your answer to two decimal places (remember, probabilities are, by definition, numbers between 0 and 1, so express your answer as a decimal).

Answer 1

The probability of drawing 2 red marbles out of 4 marbles, with a proportion of red marbles being 0.22, is approximately 0.30.

Step-by-step explanation:

The probability of drawing 2 red marbles out of 4 marbles, with a proportion of red marbles being 0.22, can be calculated using the binomial distribution formula. The formula for the probability of exactly k successes in n independent Bernoulli trials is: P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where C(n,k) is the binomial coefficient.

In this case, n=4 (number of trials), k=2 (number of successes), and p=0.22 (proportion of red marbles). Plugging in these values, we can calculate the probability as follows:

P(X=2) = C(4,2) * 0.22^2 * (1-0.22)^(4-2)

P(X=2) = 6 * 0.22^2 * 0.78^2

P(X=2) ≈ 0.3012

Therefore, the probability of drawing 2 red marbles out of 4 marbles, with a proportion of red marbles being 0.22, is approximately 0.30.