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C1. (a) (i) State the addition rule for two general events E and F. (ii) Using the addition rule for two general events, and clearly stating any set iden- tities, prove that Pr(AUBUC) = Pr(A) + Pr(B) + Pr(C) - Pr(An B) - Pr(BNC) - Pr(ANC) + Pr(An BNC) where A, B and C are three general events associated with the sample space . (iii) Now suppose that events A and B are independent, that events A and Care independent and that events B and C are mutually exclusive. How does the right-hand side of the above equation simplify to depend on Pr(A), Pr(B) and Pr(C) only?

C1. (a) (i) State the addition rule for two general events E and F. (ii) Using the addition rule for two general events, and clearly stating any set iden- tities, prove that Pr(AUBUC) = Pr(A) + Pr(B) + Pr(C) - Pr(An B) - Pr(BNC) - Pr(ANC) + Pr(An BNC) where A, B and C are three general events associated with the sample space . (iii) Now suppose that events A and B are independent, that events A and Care independent and that events B and C are mutually exclusive. How does the right-hand side of the above equation simplify to depend on Pr(A), Pr(B) and Pr(C) only?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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could you pleaae do all parts ,with explanations 

C1. (a) (i) State the addition rule for two general events E and F. (ii) Using the addition rule for two general events, and clearly stating any set iden- tities, prove that Pr(AUBUC) = Pr(A) + Pr(B) + Pr(C) - Pr(An B) - Pr(BNC) - Pr(ANC) + Pr(An BnC) where A, B and C are three general events associated with the sample space N. (iii) Now suppose that events A and B are independent, that events A and Care independent and that events B and C are mutually exclusive. How does the right-hand side of the above equation simplify to depend on Pr(A), Pr(B) and Pr(C) only? (b) There are four basic human blood groups: A, B, AB and O. Of UK blood donors, about 37% are type A, 10% are type B, 3% are type AB and 50% are type O. (i) What is the probability that there are exactly 4 people of blood type A in a random sample of 10 donors? (ii) Suppose that a new sample survey is being designed. What sample size is needed so that there is a probability of at least 0.98 of there being at least one person of blood type A in the sample (iii) Now consider a random sample of size 100. Using a suitable approximation, what is the approximate probability that at most half the sample have blood type O? (iv) Write down R commands to evaluate each of the two probabilities in parts (b)(i) and (b)(iii) above.
Transcribed Image Text:C1. (a) (i) State the addition rule for two general events E and F. (ii) Using the addition rule for two general events, and clearly stating any set iden- tities, prove that Pr(AUBUC) = Pr(A) + Pr(B) + Pr(C) - Pr(An B) - Pr(BNC) - Pr(ANC) + Pr(An BnC) where A, B and C are three general events associated with the sample space N. (iii) Now suppose that events A and B are independent, that events A and Care independent and that events B and C are mutually exclusive. How does the right-hand side of the above equation simplify to depend on Pr(A), Pr(B) and Pr(C) only? (b) There are four basic human blood groups: A, B, AB and O. Of UK blood donors, about 37% are type A, 10% are type B, 3% are type AB and 50% are type O. (i) What is the probability that there are exactly 4 people of blood type A in a random sample of 10 donors? (ii) Suppose that a new sample survey is being designed. What sample size is needed so that there is a probability of at least 0.98 of there being at least one person of blood type A in the sample (iii) Now consider a random sample of size 100. Using a suitable approximation, what is the approximate probability that at most half the sample have blood type O? (iv) Write down R commands to evaluate each of the two probabilities in parts (b)(i) and (b)(iii) above.
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