Question

How many different license plates are possible if two characters are unknown but we know that they are numerals? A random sample of 4 claims are selected from a lot of 12 that has 3 nonconforming units. Using the hypergeometric distribution, what is the probability that the sample will contain exactly 0 nonconforming units? 1 non conforming unit? A steady stream of invoices has a nonconformity of .03. What is the probability of obtaining 2 nonconforming units from a sample of 20 using the binomial distribution.

Answer 1

The probability of selecting a certain number of nonconforming units in a sample can be found using the hypergeometric distribution. For the given questions, we can use the hypergeometric distribution to find the probabilities of selecting 0 nonconforming units, 1 nonconforming unit, and the binomial distribution to find the probability of obtaining 2 nonconforming units in a sample.

Step-by-step explanation:

Hypergeometric Distribution:

To find the probability of selecting a certain number of nonconforming units in a sample, we can use the hypergeometric distribution. The hypergeometric distribution is used when sampling without replacement from a population with two groups. In this case, the two groups are conforming and nonconforming units. We can find the probability of selecting a specific number of nonconforming units by dividing the number of ways to select that many nonconforming units by the total number of possible samples.

1) Probability of 0 nonconforming units:

To find the probability of selecting exactly 0 nonconforming units, we can use the formula:

P(X = 0) = (C(3, 0) * C(9, 4)) / C(12, 4)

Let's solve this expression step by step:

C(3, 0) = 1
C(9, 4) = 126
C(12, 4) = 495

P(X = 0) = (1 * 126) / 495 = 0.2545

So, the probability of selecting exactly 0 nonconforming units is approximately 0.2545.

2) Probability of 1 nonconforming unit:

To find the probability of selecting exactly 1 nonconforming unit, we can use the formula:

P(X = 1) = (C(3, 1) * C(9, 3)) / C(12, 4)

Let's solve this expression step by step:

C(3, 1) = 3
C(9, 3) = 84
C(12, 4) = 495

P(X = 1) = (3 * 84) / 495 = 0.5091

So, the probability of selecting exactly 1 nonconforming unit is approximately 0.5091.

Binomial Distribution:

To find the probability of obtaining a certain number of nonconforming units in a sample using the binomial distribution, we need to know the probability of a nonconforming unit in each individual unit. In this case, we are given that the nonconformity rate is 0.03. We can then use the formula for the binomial distribution to find the probability of obtaining a specific number of nonconforming units in a sample.

3) Probability of 2 nonconforming units:

To find the probability of obtaining exactly 2 nonconforming units in a sample of 20, we can use the formula:

P(X = 2) = C(20, 2) * (0.03)^2 * (1-0.03)^(20-2)

Let's solve this expression step by step:

C(20, 2) = 190

P(X = 2) = 190 * (0.03)^2 * (1-0.03)^(20-2) = 0.2706

So, the probability of obtaining exactly 2 nonconforming units in a sample of 20 is approximately 0.2706.