5. A bag contains 4 red and 6 blue marbles. A marble is chosen at random, its colour noted and then returned to the bag. This procedure happens eleven times.
What is the probability that exactly 4 red marbles are drawn? What is the most likely number of blue marbles drawn?
Answer
\(\boxed{\text{The probability of drawing exactly 4 red marbles is approximately 0.2365, and the most likely number of blue marbles drawn is 7.}}\)
Step 1 ：Let n be the number of trials (11), k be the number of successes (4 red marbles), p be the probability of success (4/10 for red marbles), and C(n, k) be the number of combinations of n items taken k at a time.
Step 2 ：Use the binomial probability formula: \(P(X = k) = C(n, k) * p^k * (1-p)^{(n-k)}\)
Step 3 ：Calculate the probability of drawing exactly 4 red marbles: \(P(X = 4) = C(11, 4) * (0.4)^4 * (1-0.4)^{(11-4)}\)
Step 4 ：\(P(X = 4) \approx 0.2365\)
Step 5 ：Calculate the probabilities for each possible number of blue marbles drawn (from 0 to 11) and find the maximum probability.
Step 6 ：Most likely number of blue marbles drawn is 7.
Step 7 ：\(\boxed{\text{The probability of drawing exactly 4 red marbles is approximately 0.2365, and the most likely number of blue marbles drawn is 7.}}\)
