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Answer:
The expected cost of the company for a 3000 tires batch is $120255
Step-by-step explanation:
Recall that given a probability of defective tires p, we can model the number of defective tires as a binomial random variable. For 3000 tires, if we have a probability p of having a defective tire, the expected number of defective tires is 3000p.
Let X be the number of defective tires. We can use the total expectation theorem, as follows: if there are [tex]A_1,\dots, A_n[/tex] events that partition the whole sample space, and we have a random variable X over the sample space, then
[tex]E(X) = p(A_1)E(X|A_1) + \dots + p(A_n)E(X|A_n)[/tex].
So, in this case, we have the following
[tex]E(X) = P(p=0\%)E(X|p=0\%)+P(p=1\%)E(X|p=1\%)+P(p=2\%)E(X|p=2\%)+P(p=3\%)E(X|p=3\%) = 0.1\cdot 3000\cdot 0 + 0.3\cdot 3000\cdot 0.01+ 0.4\cdot 3000\cdot 0.02+ 0.2\cdot 3000\cdot 0.03 = 51[/tex].
Let Y be the number non defective tires. then X+Y = 3000. So Y = 3000-X. Then E(Y) = 3000-E(X). Then, E(Y) = 2949.
Finally, note that the cost of the batch would be 40Y+45X. Then
[tex]E(40Y+45X) = 40E(Y)+45E(X) = 40\cdot 2949+45\cdot 51=120255[/tex]
The expected cost of replacement of the company for a 3000 tiers batch is $120,225.
What is probability?
Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.
A company manufactures tires at a cost of $40 each.
The following are probabilities of defective tires in a given production run:
- The probability of 0% defective tires is 10%.
- The probability of 1% defective tires is 30%.
- The probability of 2% defective tires is 40%.
- The probability of 3% defective tires is 20%.
When a defective tire is detected, it must be removed from the assembly line and replaced.
This process adds an extra $5 to the cost of the replaced tire.
Let x be the number of defective tires. We can use the total expectation theorem as follow
If there are A₁, A₂,....., [tex]\rm A_n[/tex] events that partition the whole sample space, and we have a random variable X over the sample space. Then
[tex]\rm E(X) = p(A_1)E(X|A_1) + ...... + p(A_n)E(X|A_n)\\\\[/tex]
So we have
[tex]\rm E(X) = P(p=0\%) E(X|_p = 0\%) + P(p=1\%) E(X|_p = 1\%) + P(p=2\%) E(X|_p = 2\%) + P(p=3\%) E(X|_p = 3\%)\\\\E(X) = 0.1*3000*0 + 0.3*3000*0.01 + 0.4*3000+0.02 + 0.2*3000*0.03\\\\E(X) = 51[/tex]
Let Y be the number of non-defective tires. Then
X + Y = 3000
E(X) + E(Y) = 3000
51 + E(Y) = 3000
E(Y) = 2949
Finally, the cost of the batch will be 45X + 40Y. Then
[tex]E(40Y + 45X) = 40E(Y) + 45E(X) = 40*2949 + 45*51 = 120255[/tex]
More about the probability link is given below.
https://brainly.com/question/795909
