1.A lot of 30 spacing washers contain 10 washers that arethicker than the target

Question

1.A lot of 30 spacing washers contain 10 washers that arethicker than the target dimension and the rest are at targetdimension. Suppose that 3 washers are selected at random,without replacement, from the lot.

a.What is the probability that all three washers are thickerthan the target?

b.What is the probability that the third washer selected isthicker than the target if the first two washers selected are notthicker than the target?

c.What is the minimum number of washers that need to be selectedso that the probability that all the washers are at targetdimension is less than 0.3?

d.What is the minimum number of washers that need to be selectedso that the probability that one or more washers are thicker thanthe target is at least 0.5?

Explanation / Answer

For the first two problems, the denominator would be C(30,3)since you are chosing 3 washers from a total of 30. a.)There are 10 washers that are thicker than the targetdimension.  You can only chose 3 of these 10washers. Thus, the probability that all three washers arethicker than the target is: C(10,3)/C(30,3)=.029557 b.)The first two washers are chosen out of the 20 washers thatare not thicker. The last washer is chosen out of the 10washers that are thicker. Thus, the probability that thefirst two washers are thicker than the target and the third washeris thicker than the target is: C(20,2)*C(10,1)/C(30,3)=.077997 c.)Now you have 20 washers that you want to select from. You can chose "n" washers from these 20. This would yieldC(20,n) combinations. The demoninator would be C(30,n) sinceyou are chosing the same number of washers. C(20,n)=20!/[(20-n)!n!] C(30,n)=30!/(30-n)!n!] The n!'s can cancel it since you're dividing the two. And 20!/(20-n)!=20*19*18*... in which there are n number ofterms. 30!/(30-n)!=30*29*28* ... in which there are n number of termsagain. So basically you're looking for: (20*19*18*...)/(30*29*28*...)<0.3 20/30=2/3=.666666666667 20*19/(30*29)=2/3*19/29=38/87= 20*19*18/(30*29*28)=2/3*19/29*14/19=.322 .322*17/27 is less than 0.3. Since we are at the 4thterm, and n is the variable we're looking for, the answer is4 minimum washers selected. d.)The probability that one or more washers is thicker = 1 -the probability that none of the washers are thicker. 0.5=1-0.5 We can attack this question in a very similar way. Onceagain, there are 20 washers from which to chose from, and so on andso forth. We are solving the same inequality for "n" except that now weare looking at 0.5, not 0.3. From the previous question, we see that by the time we arrivedat the 2nd term in the sequence, it was already less than .5 Thus, the minimum number is 1washer.

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