the owner of supereb coffee wants to buy 100 bags of coffee from Great Aroma. They take a sample of 10 bags and agree to buy the consigment if, at most , 1 bag in the sample fails to meet requirements. Great Aroma knows that 10 % of the total consigment will not meet their requirements. What is the probability that they will buy the consigment?
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I took 100 and - 10 bags because they were taking them as a sample= 90 then there is one bag that doesnt meet requirements to it would be 89. but once i got here if this is even right i dnt know what to do
To find the probability that Great Aroma will buy the consignment, we need to determine the probability of having 1 or fewer bags in the sample that fail to meet the requirements.
First, let's calculate the number of bags in the entire consignment that won't meet the requirements. We know that 10% of the total consignment won't meet the requirements, so if the owner wants to buy 100 bags, then there will be 0.1 * 100 = 10 bags that fail to meet the requirements.
Next, we need to calculate the probability of having 1 or fewer bags in the sample that fail to meet the requirements. To do this, we can use the binomial probability formula:
P(X ≤ 1) = P(X = 0) + P(X = 1)
where X is the number of bags in the sample that fail to meet the requirements.
The formula for calculating the probability of exactly X successes (or failures) in a sample of size n is:
P(X) = C(n, X) * p^X * q^(n-X)
where C(n, X) is the number of combinations of n items taken X at a time (binomial coefficient), p is the probability of success (bag meets requirements), and q is the probability of failure (bag doesn't meet requirements).
In this case, since the owner wants to buy the consignment if at most 1 bag fails to meet the requirements, we can substitute n = 10 (sample size) and p = 0.9 (probability of success) into the formula.
P(X = 0) = C(10, 0) * 0.9^0 * 0.1^10
P(X = 1) = C(10, 1) * 0.9^1 * 0.1^9
To calculate the binomial coefficients, we can use the formula:
C(n, X) = n! / (X! * (n - X)!)
Substituting the values:
C(10, 0) = 10! / (0! * (10 - 0)!) = 1
C(10, 1) = 10! / (1! * (10 - 1)!) = 10
Now we can calculate the probabilities:
P(X = 0) = 1 * 0.9^0 * 0.1^10 = 0.1^10 = 0.0000000001
P(X = 1) = 10 * 0.9^1 * 0.1^9 = 10 * 0.9 * 0.1^9 = 0.000000009
Finally, we add the probabilities together to find the probability that they will buy the consignment:
P(X ≤ 1) = P(X = 0) + P(X = 1) = 0.0000000001 + 0.000000009 = 0.0000000091
Therefore, the probability that Great Aroma will buy the consignment is approximately 0.0000000091.