Question

An assembly consists of three mechanical components. Suppose that the probabilities that the first, second, and third components meet specifications are 0.95, 0.98, and 0.99, respectively. Assume that the components are independent. Determine the probability mass function of the number of components in the assembly that meet specifications. 3.1.18 The distribution of the time until a Web

Answer 1

The probability mass function of the number of components that meet specifications is calculated by considering all possible combinations of the components' individual probabilities for meeting specifications, given they are independent.

Step-by-step explanation:

To calculate the probability mass function for the number of components that meet specifications in the assembly, we have to consider all combinations of the components meeting or not meeting specifications. With three independent components and their probabilities for meeting specifications being 0.95, 0.98, and 0.99, we can determine the probabilities of having 0, 1, 2, or all 3 components meeting the specifications.

Let's define X as the number of components meeting specifications, then X can take on the value of 0, 1, 2, or 3.

• P(X=0) would be the probability that none of the components meet specifications. This is found by multiplying the probabilities that each one fails: (1-0.95)*(1-0.98)*(1-0.99).
• P(X=1) would be the probability that exactly one component meets the specifications. This requires summing up the probabilities for each component being the one that meets specifications while the others do not.
• P(X=2) would involve calculating the sum of the probabilities for any two components meeting specifications while one does not.
• Lastly, P(X=3) is simply the probability that all components meet specifications, which is the product of each of their individual probabilities: 0.95*0.98*0.99.

To represent the distribution, we create a probability distribution histogram where the x-axis represents the number of components meeting specifications (X) and the y-axis represents the probability of that scenario occurring.

Answer 2

P(X=0)=1x10^-5

P(X=1)=1.67x10^-3

P(X=2)=0.0766

P(X=3)=0.9217

Step-by-step explanation:

X="The number of components that meet specifications"

A="The 1st component meet the specification"

B="The 2nd component meet the specification"

C="The 3rd component meet the specification"

The events are independents

P(A)=0.95

P(B)=0.98

P(C)=0.99

P(X=0)=P(A'∩B'∩C')=P(A')P(B')P(C')=(1-P(A))(1-P(B))(1-P(C))=

(1-0.95)(1-0.98)(1-0.99)=0.05x0.02x0.01=1x10^-5

P(X=0)=1x10^-5

P(X=1)=P(A∩B'∩C')+P(A'∩B∩C')+P(A'∩B'∩C)

P(X=1)=P(A)P(B')P(C')+P(A')P(B)P(C')+P(A')P(B')P(C)

P(X=1)=P(A)(1-P(B))(1-P(C))+(1-P(A))P(B)(1-P(C))+(1-P(A))(1-P(B))P(C)

P(X=1)=(0.95)(1-0.98)(1-0.99)+(1-0.95)(0.98)(1-0.99)+(1-0.95)(1-0.98)(0.99) P(X=1)=1.67x10^-3

P(X=2)=P(A'∩B∩C)+P(A∩B'∩C)+P(A∩B∩C')

P(X=2)=P(A')P(B)P(C)+P(A)P(B')P(C)+P(A)P(B)P(C')

P(X=2)=(1-P(A))P(B)P(C))+P(A)(1-P(B))P(C)+P(A)P(B)(1-P(C))

P(X=2)=(0.05)(0.98)(0.99)+(0.95)(0.02)(0.99)+(0.95)(0.98)(0.01)

P(X=2)=0.0766

P(X=3)=P(A∩B∩C)=P(A)P(B)P(C)=P(A)P(B)P(C)= (0.95)(0.98)(0.99)=

P(X=3)=0.9217