Question

What is Bayes' theorem?

Answer 1

Detailed explanation is given below.

Explanation:

The probability that we calculate are of two types.

1. the a - priori probability.
2. The a- posteriori probability.
   As the name suggests, a-priori probability is a probability calculated before the happening of an event.
   a-posteriori probability is a probability calculated after the happening of an event.
   The posteriori probabilities are known as Bayesian probabilities.
   This is because, this method was first suggested by Bayes.
   The answer is continued further. Check my next answer also.

Answer 2

The answer continues from my earlier answer.

Explanation:

Imagine this situation.

I have three children who take turn to cleaning the dish in the kitchen.
Let's say that `C_1` represents the event the first child cleans.
Let `C_2` represent the event that the second child cleans.
Let `C_3` represent the event the third child cleans
Let B be the event that a dish is broken,
Then Pr`[B/C_1]` represents the conditional probability that the dish is broken by the first child. This is an apriori probability because we calculate the probability before the dish is broken.
But Pr`[C_1/B]` is a-posteriori probability since we now calculate the probability after the event, the dish is broken.
The answer is continued.

Answer 3

The answer continues further.

Explanation:

Now let's form a proper question.

The probability that one of my three children `C_1`, `C_2`, or `C_3` wash the dish in the kitchen on any given day is 0.4, 0.25 and 0.35 respectively.
The probability that `C_1` breaks a dish is given by Pr `[B/C_1]` is 0.3
The probability that `C_2` breaks the dish is Pr `[B/C_2]` is 0.6.
The probability that `C_3` breaks the dish is Pr `[B/C_3]` is 0,7.
One day, I hear the sound of a dish being broken.
What is the probability that cleaning was being done by `C_2` that day?
The answer continues.

Answer 4

The answer continues.

Explanation:

In terms of conditional probability, we must calculate Pr `[C_1/B]`.
By BAYES' theorem, it is obtained as follows.
Pr `[C_2/B]` = `[(Pr(C_2),Pr(B/C_2))/{Pr(C_1)Pr(B/C_1)+Pr(C_2)Pr(B/C_2)+Pr(C_3)Pr(B/C_3)}]`.
Pr`[B/C_2]` = `[((1/3).(0.6))/{(1/3)(0.3)+(1/3).(0.6)+(1/3).(0.7)}]`