Cartons of milk from a particular supermarket are advertised as containing 1 litre of milk,but in fact the volume of the milk in a cartoon is normally distributed with mean 1012 ml and standard deviation 5 ml.
i.Find the probability that exactly 3 cartons in a sample of 10 cartons contain more than 1012 ml.
well, what was your answer?
You are working with the z-score : )
Yes I am.Tell me how to do.
Thanks.
0.1171
To find the probability that exactly 3 cartons in a sample of 10 contain more than 1012 ml, we can use the concept of the binomial distribution.
The binomial distribution is used to calculate the probability of having a specific number of successes in a fixed number of independent Bernoulli trials (where each trial has only two possible outcomes - success or failure) with a known probability of success.
In this case, we want to find the probability of having exactly 3 successes (cartons with more than 1012 ml) out of 10 trials (10 cartons). Let's denote the probability of a carton containing more than 1012 ml as p.
First, we need to calculate the probability of success, which is the probability of a carton containing more than 1012 ml.
Using the z-score formula, we can standardize the value of 1012 ml with the mean and standard deviation provided:
z = (x - mean) / standard deviation
z = (1012 - 1012) / 5
z = 0
Next, we can use a standard normal distribution table or a calculator to find the probability associated with a z-score of 0. This probability represents the area under the standard normal curve to the right of the z-score.
Since we're interested in the probability of having more than 1012 ml, we calculate 1 minus the probability we obtained (to account for the right-tail).
So, the probability of a single carton containing more than 1012 ml can be obtained as follows:
p = 1 - P(Z < 0)
Now that we have the probability of success for a single trial, we can proceed to calculate the probability of exactly 3 successes in a sample of 10 trials.
Using the binomial probability formula, the probability of exactly k successes in n trials with a probability of success p is given by:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case, n = 10 (number of trials), k = 3 (number of successes), and p is the probability of a single carton containing more than 1012 ml.
P(X = 3) = (10 choose 3) * p^3 * (1 - p)^(10 - 3)
Substituting the values we obtained:
P(X = 3) = (10 choose 3) * p^3 * (1 - p)^7
Using the combination formula, (10 choose 3) = 120, we can calculate the probability:
P(X = 3) = 120 * p^3 * (1 - p)^7
Finally, substituting the value of p (obtained earlier) into the equation, we can find the probability that exactly 3 cartons in a sample of 10 cartons contain more than 1012 ml.
First of all find the probabilty of a carton containing more than 1012 ml
using the webpage I gave you earlier.
Then use the Binomial distribution to find the prob(exactly 3 of 10) ....