Explain the following proposition with example: “Probability provides a way of summarizing the uncertainty that comes from our laziness and ignorance”.

If ?(? ∪ ?) = 0.64 and Event A and B are mutually exclusive, which of the following could be possible probabilities for A and B? a. ?(?)=0.64 , ?(?)=0.64 b. ?(?)=0.8 , ?(?)=0.8 c. ?(?)=0.42 , ?(?)=0.22 d. ?(?)=0.8 , ?(?)=0.16

The following problem includes 5 parts.First, state the definition of conditional probability. Second, use the definition of conditional probability and derived rules to prove Bayes’ Theorem. Third, state Bayes’ Rule. Fourth, explain the difference between Bayes’ Theorem and Bayes’ Rule. Fifth, construct and solve a probability reasoning problem that uses Bayes’ Theorem.

Suppose we know that the probabilities P(A) = .5, P(B|A) = .3 and P(A or B) =.3. What is the value for P(A and B)? What is the value for P(B)?

Four people are running for class​ president: Liz,​ Fred, Sue, and Tom. The probabilities of​ Fred, Sue, and Tom winning are 0.26​, 0.28​, and 0.29​, respectively.   ​(a) What is the probability of Liz​ winning? ​(b) What is the probability that Fred or Tom​ wins? ​(c) What is the probability that Tom​ loses?   ​(a) The probability of Liz winning is nothing. ​(Type an integer or a​ decimal.)

Which of the following choices correctly describe the probability of picking a dog that is a Yorkie given the dog prefers to eat chicken? O P(Yorkie|chicken) O P(chicken and Yorkie) O P(chicken | Yorkie) O P(Yorkie or chicken) Which of the following choices correctly describe the probability of picking a dog that is a Shih Tsu and the dog prefers to eat cheese? O P(Shih Tsu | cheese) O P(Shih Tsu or cheese) O P(Shih Tsu and cheese) O P(cheese | Shih Tsu)

Researchers surveyed 100 students on which superpower they would most like to have. This two-way table displays data for the sample of students who responded to the survey: Superpower Male Female TOTAL Fly 26 12 38 Invisibility 12 32 44 Other 10 8. 18 TOTAL 48 52 100

When playing games of chance like roulette, people often say that a particular outcome is due, implying that one income is more likely because it has not happened in a while. This is fallacy because in these examples of outcome of each trial is: a. independent b. dependent c. subjective d. interrelative

two way table                  Pathology     Pediatrics      Psychiatry Male        2950               1380              270          4600 Female    1550               1320            1530          4400                              4500               2700            1800          9000     Find each of the following conditional probabilities   a) Are the events E = 'Female' and F = 'Pathology' dependent or independent?  Explain why.       b) Are the events F = 'Male' and G = 'Pediatrics' dependent or independent?  Explain why.

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