Question 1Given the following information about events A, B, and C, determine which pairs of events, if any, are independent and which pairs are mutually exclusive.P(A)=0.26 , P(B)=0.5 , P(C)=0.45 . P(A|B)=0.26 , P(B|C)= 0 , P(C|A)=0.26Select all correct answers.Select all that apply:B and C are independentA and C are mutually exclusiveA and B are independentA and C are independentB and C are mutually exclusiveA and B are mutually exclusiveQuestion 2Let E be the event that a randomly chosen person exercises. Let D be the event that a randomly chosen person is on a diet. Identify the answer which expresses the following with correct notation: Of all the people who exercise, the probability that a randomly chosen person is on a diet.Select the correct answer below:P(D) AND P(E)P(E AND D)P(E|D)P(D|E)

Answer with Step-by-step explanation:1.We are given that  three events  A, B and C.P(A)=0.26P(B)=0.5P(C)=0.45P(A/B)=0.26P(B/C)=0P(C/A)=0.26When two events A and B are independent then [tex]P(A\cap B)=P(A)\cdot P(B)[/tex]If two events are mutually exclusive then [tex]P(A\cap B)=0[/tex]We know that [tex]P(A/B)=\frac{P(A\cap B)}{P(B)}[/tex][tex]P(A\cap B)=P(A/B)\times P(B)[/tex][tex]p(A\cap B)=0.26\times 0.5=0.13[/tex][tex]P(A)\times P(B)=0.26\times 0.5=0.13[/tex]Hence, [tex]P(A\cap B)=P(A)\cdot P(B)[/tex]Therefore, event A and B are independent.[tex]P(B\cap C)=0\times 0.45=0[/tex]Therefore, events B and C are mutually exclusive.[tex]P(A\cap C)=0.26\times 0.26=0.0676[/tex][tex]P(A)\times P(C)=0.26\times 0.45=0.117[/tex][tex]P(A\cap C)\neq P(A)\cdot P(C)[/tex]Hence, event A and C are neither independent nor mutually exclusive.Answer: A and B are independentB and C are mutually exclusive.2.Let E be the event that  randomly chosen person exercises and D be the event that a randomly chosen person is on a diet.According to question We have to find P(D/E).Answer : P(D/E)