Answer:
Step-by-step explanation:
The persons who have diseases = 19%
The persons who do not have diseases = 81%
Probability for the test showing positive = Prob of actual disease and shows positive + Prob of no disease but test shows wrong
= ![0.7*0.19+0.04*0.81\\=0.133+0.0324\\=0.1654](https://tex.z-dn.net/?f=0.7%2A0.19%2B0.04%2A0.81%5C%5C%3D0.133%2B0.0324%5C%5C%3D0.1654)
Prob that the person actually has the disease given that the test administered to an individual is positive= ![\frac{0.133}{0.1654} \\=0.0804](https://tex.z-dn.net/?f=%5Cfrac%7B0.133%7D%7B0.1654%7D%20%5C%5C%3D0.0804)
30) A B C total
2x 7x x 10x
P(A) = 0.2, P(B) = 0.7, P(C) = 0.1
Let D be the event that the product is defective
P(D/A) = 0.032, P(D/B) = 0.034 and P(D/C) = 0.056
Required probability =P(C/D)
= ![\frac{P(D/C) P(C)}{P(D/A) P(a)+P(D/B) P(B)+P(D/C) P(C)}](https://tex.z-dn.net/?f=%5Cfrac%7BP%28D%2FC%29%20P%28C%29%7D%7BP%28D%2FA%29%20P%28a%29%2BP%28D%2FB%29%20P%28B%29%2BP%28D%2FC%29%20P%28C%29%7D)
by using Bayes theorem for conditional probability
= ![\frac{0.1*0.056}{0.2*0.032+0.7*0.034+0.1*0.056} \\=\frac{0.0056}{0.0064+0.0238+0.0056} \\\\=0.156425](https://tex.z-dn.net/?f=%5Cfrac%7B0.1%2A0.056%7D%7B0.2%2A0.032%2B0.7%2A0.034%2B0.1%2A0.056%7D%20%5C%5C%3D%5Cfrac%7B0.0056%7D%7B0.0064%2B0.0238%2B0.0056%7D%20%5C%5C%5C%5C%3D0.156425)
32) 7 digit phone numbers with repitition including 0 will be
= ![10^7](https://tex.z-dn.net/?f=10%5E7)
Numbers which have atleast one 7 = total number - numbers which do not have a single 7
= ![10^7 -9^7](https://tex.z-dn.net/?f=10%5E7%20-9%5E7)
the probability that a 7-digit phone number contains at least one 7
=![\frac{10^7-9^7}{10^7}](https://tex.z-dn.net/?f=%5Cfrac%7B10%5E7-9%5E7%7D%7B10%5E7%7D)