USF Information Science Blog

A1. Event A
The probability of Event A is (10+20)/90 = 0.3333 (1/3)

A2. Event B?
The probability of Event B is (10+20)/90 = 0.3333 (1/3)

A3. Event A or B
To find the probability of A or B, we use the formula:
P(A∪B)=P(A)+P(B)−P(A∩B)

P(A∩B) is equal to the probability of Event A and B
The probability of Event A and B is 10/90 = 0.1111 (1/9)

The probability of Event A OR B is (1/3)+(1/3)-(1/9)=0.5556 (5/9)

A4. P(A∪B)  = P(A) + P(B)
The formula P(A∪B)  = P(A) + P(B) only holds true when events A and B are mutually exclusive, meaning they cannot happen at the same time. For this chart, A and B are not mutually exclusive, as they can occur concurrently.

However, if events A and B are not mutually exclusive, the formula becomes: P(A∪B)=P(A)+P(B)−P(A∩B) (which was used in A3)

When events A and B are not mutually exclusive, they may both occur at the same time. Therefore, the intersection P(A∩B) (the probability that both A and B happen together) is subtracted to avoid double-counting.

B. Applying Bayes’ Theorem 

B1. Is this answer True or False.
This conclusion is true

B2. Please explain why?

P( A1 ) = 5/365 =0.0136985 [It rains 5 days out of the year.]
P( A2 ) = 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.]

We are trying to find the probability of there being rain on Janes weddings given the weatherman predicts rain: P(A1​∣B). Bayes Theorem can be written as such:

N: P(A1​)⋅P(B∣A1​)​
D: P(A1​)⋅P(B∣A1​)+P(A2​)⋅P(B∣A2​)

N: P(A1​)⋅P(B∣A1​) = 0.0136985×0.9 = 0.0123287
D: P(A1​)⋅P(B∣A1​)+P(A2​)⋅P(B∣A2​) = 0.0123287+0.09863014 = 0.11095879

P(A1​∣B) = 0.11095879 / 0.01232865​ = 0.1111 , or, 11.11%

The probability that it rains on any given day in the desert is very low: only 5/365≈1.37%. This means that, despite the accuracy of the weatherman’s prediction rate, the base rate of rain is still quite low. Since it rarely rains, most of the days when the weatherman predicts rain, it actually won’t rain. This is why the final probability is much lower than the weatherman’s accuracy for predicting rain. In conclusion, the result is True because the calculation is based on accurate probabilities and follows Bayes’ Theorem correctly, but the seemingly low probability reflects the rarity of rain in the desert, despite the weatherman’s forecast.

C. Last assignment from our textbook, pp. 55 Exercise # 2.3.

This is a binomial probability problem, where:

• size=10 (the number of patients),
• prob=0.80 (the probability of no complications for each patient),
• x=10 (the number of patients with no complications).

[1] 0.1073742

The probability of of operating on 10 patients successfully with the tradtional method is 10.73%