Module #4 Assignment

Probability theory

 A. Based on Table 1 What is the probability of:

	B	B1
A	10	20
A1	20	40

Total = 10+20+20+40=90

A1. Event A

P(A) = Total for A / Total Number of events = 30/90 = 1/3​

Total for A = 10 + 20 = 30

A2. Event B?

P(B) = Total for B/ Total Number of events = 30/90 = 1/3

Total for B = 10 + 20 = 30

A3. Event A or B

P(A and B)= 10/90 = 1/9

P(A or B)=P(A) + P(B) − P(A and B) = 1/3 + 1/3 − 1/9

Common Denominator:

P(A or B) = 3/9 + 3/9 - 1/9 = 5/9

A4. P(A or B) = P(A) + P(B)

P(A or B)  ≠ P(A)+P(B)

P(A or B) = 5/9

P(A) + P(B) = 1/3 + 1/3 = 2/3

5/9  ≠  2/3  2

P(A or B) does not equal P(A)+P(B). P(A or B) is the probability that either A or B or both happen. P(A)+P(B) is if you add the probabilities of A and B. P(A)+P(B) counts the events that occur under A and B twice. P(A or B) requires subtracting P(A and B) as demonstrated above. 

B. Applying Bayes' Theorem 
Jane is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time.

What is the probability that it will rain on the day of Jane's wedding? 

Event A1. It rains on Jane's wedding.

Event A2. It does not rain on Marie's wedding.

Event B. The weatherman predicts rain.

P(A1) = 5/365 = 0.0136985 [It rains 5 days out of the year.]

P(B) = 360/365 = 0.9863014 [It does not rain 360 days out of the year.]

P( B | A1 ) = 0.9 [When it rains, the weatherman predicts rain 90% of the time.]

P( B | A2 ) = 0.1 [When it does not rain, the weatherman predicts rain 10% of the time.]

P( A1 | B ) = ?

Baye's Theorem - 

P( A1 | B ) = P( A1 ) P( B | A1 )

P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2 )

P( A1 | B ) = (0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.1) ]

P( A1 | B ) = 0.111

11.1%

Even when the weatherman predicts rain, it only rains only about 11% of the time. Despite the weatherman's gloomy prediction, there is a good chance that Marie will not get rained on at her wedding.

B1. Is this answer True or False.

True

B2. Please explain why?

There is only a 11.1% chance of raining on the day of the wedding. Even though the weatherman has an almost accurate prediction rate, the chance for rain is still very low. The rare chance of it raining outweighs the weatherman's prediction. The weatherman's prediction is likely a false positive which in turn results in a 11.1% chance of it raining. 

C. Last assignment from our textbook, pp. 55 Exercise # 2.3.
For a disease known to have a postoperative complication frequency of 20%, a surgeon suggests a new procedure. She/he tests it on 10 patients and found there are not complications. What is the probability of operating on 10 patients successfully with the traditional method?

> dbinom(10, size=10, prob=0.80)

[1] 0.1073742

10.74%

Recap:

This week's assignment gave an extensive overview of probability. I gained a better understanding of conditional probability as well as independent events and overlapping events. Using the dbinom function, I can calculate different scenarios much faster and more efficiently. As I continue along with this course, I hope to integrate the concepts that I learned from this week into more complex statistical problems.