Module #4 Probability Theory

 A1. P(A) = P(B) + P(B1) = 10 + 20 = 30 

A2. P(B) = P(A) + P(A1) = 10 + 20 = 30 

A3. P(A or B) = P(A) + P(B) - P(A and B) 

       P(A and B) = 10 

       P (A or B) = P(A) + P(B) - P(A and B) = 30 + 30 - 10 = 50 

       Therefore, the probability of Event A or B is 50%. 

A4. P(A or B) = P(A) + P(B) 

       P(A or B) = 30 + 30 = 60

       Therefore, P(A or B) is 60%.  


B1. True

B2. 

The answer is true because it is based on the application of Bayes' theorem. In this case, the question is asking for the probability that it will rain on the day of Jane's wedding (Event A1) given the weatherman's prediction of rain (Event B). 

The calculation correctly takes into account the probabilities of it raining (P(A1)) and not raining (P(A2)), as well as the conditional probabilities of the weatherman correctly predicting rain when it actually rains (P(B|A1)) and incorrectly when it doesn't rain (P(B|A2)). 

The result is 11% meaning, there's only an 11% chance that it will actually rain on Jane's wedding day. 

C. 

Probability_success <- dbinom(10, size = 10, prob = 0.2) 

Probability_success 

[1] 0.1073742 

So, the probability of operating on 10 patients successfully with the traditional method is approximately 10.74% 

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