Let's tackle each question one by one:
Question 1:
1.1. The precise random variable required to solve this problem using the Binomial distribution is the number of cartons (out of 12) filled with less than 750 milliliters of milk.
1.2. Two conditions necessary for a random variable to be modeled using a Binomial distribution are:
a) There must be a fixed number of trials (in this case, filling 12 cartons).
b) Each trial must be independent of the others.
c) There are only two possible outcomes for each trial (success or failure).
d) The probability of success (p) is the same for each trial.
1.3. To find the probability that exactly two cartons are filled with less than 750 milliliters of milk, you can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
n is the number of trials (12 cartons).
k is the number of successful trials (2 cartons).
p is the probability of success (1/8, because 1 in 8 cartons has less than 750ml).
P(X = 2) = (12 choose 2) * (1/8)^2 * (7/8)^(12 - 2)
Calculating this will give you the probability.
1.4. To find the probability that none of the cartons are filled with less than 750 milliliters of milk, you can use the same formula, but this time k would be 0:
P(X = 0) = (12 choose 0) * (1/8)^0 * (7/8)^12
Calculating this will give you the probability.
1.5. To find the probability that the machine will not need recalibration (i.e., more than 3 cartons in the sample have less than 750 milliliters of milk), you can find the complement of the probability that it will need recalibration (3 or fewer cartons with less than 750ml). Use the binomial probability formula for this:
P(X > 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]
Calculate the values as per the previous calculations and subtract them from 1.
Question 2:
2.1. To find the probability that a randomly selected carton that has gone through the new machine is overfull by 4 milliliters, you can use the z-score formula for a normal distribution:
Z = (X - μ) / σ
Where:
X is the value you want to find the probability for (in this case, 1004 milliliters).
μ is the mean (1,000 milliliters).
σ is the standard deviation (2.5 milliliters).
Calculate the z-score, and then use a standard normal distribution table or calculator to find the probability associated with that z-score.
2.2. To find the number of cartons expected to be sent out to the local supermarkets, you'll need to find the probability that a carton falls within the 6-milliliter range around 1 liter. This range is from 994 milliliters to 1,006 milliliters. Calculate the z-scores for these values, find the corresponding probabilities, and then multiply the total number of cartons (5000) by this probability to get the expected number of cartons within the specified range.
Question 3:
3.1. To find the probability that the machine will fill more than 4 cartons in a randomly chosen 10-second period, you can use the Poisson distribution. The Poisson distribution models the number of events that occur in a fixed interval of time. In this case, you have 3 cartons filled every 10 seconds. You can calculate the probability of filling more than 4 cartons in a 10-second period using the Poisson distribution formula.
3.2. To find the probability that the machine will fill exactly 20 cartons in a randomly chosen minute, you can again use the Poisson distribution. You have 3 cartons filled every 10 seconds, so in a minute (60 seconds), there are 6 intervals of 10 seconds. Calculate the probability of filling exactly 20 cartons in 6 intervals using the Poisson distribution formula.