A store owner complains that, on the average, 3 bottles per case of a certain brand of softdrink are underfilled. Assuming this is correct, find the probability that a randomly chosen case of this drink will contain at least 4 underfilled bottles.

To find the probability that a randomly chosen case of this drink will contain at least 4 underfilled bottles, we need to use the binomial distribution.

The binomial distribution is used when we have a fixed number of independent trials (in this case, the bottles in the case) and each trial has only two possible outcomes (underfilled or not underfilled).

Let's break down the problem:

1. Identify the parameters:
- Number of trials (n): The number of bottles in a case (assuming it's constant for all cases).
- Probability of success (p): The probability that a single bottle is underfilled.
- Number of underfilled bottles we want to find the probability for (k): 4 or more.

2. Calculate the probability of success (p):
Since the average number of underfilled bottles per case is given as 3, we can divide this by the total number of bottles in a case to get the probability of a single bottle being underfilled.
p = 3/n

3. Calculate the probability of failure (q):
The probability of a single bottle not being underfilled is the complement of the probability of success.
q = 1 - p

4. Calculate the probability of getting at least 4 underfilled bottles:
We need to calculate the probability of getting exactly 4 underfilled bottles, exactly 5 underfilled bottles, and so on, up to n underfilled bottles. Then we sum up these probabilities.

P(X >= k) = P(X = k) + P(X = k+1) + ... + P(X = n)

P(X = k) = C(n, k) * p^k * q^(n-k)
where C(n, k) is the binomial coefficient (n choose k), calculated as n! / (k! * (n-k)!)

5. Substitute the values and calculate:
Let's assume a case contains 12 bottles.

p = 3/12 = 1/4
q = 1 - p = 1 - 1/4 = 3/4

P(X >= 4) = P(X = 4) + P(X = 5) + ... + P(X = 12)

P(X = 4) = C(12, 4) * (1/4)^4 * (3/4)^8
P(X = 5) = C(12, 5) * (1/4)^5 * (3/4)^7

Continue calculating P(X = k) for k = 4 to 12, and sum up these probabilities to get the final answer.

By following these steps and substituting the values, you can calculate the probability that a randomly chosen case of this drink will contain at least 4 underfilled bottles.

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