In the last article, we discussed Baye’s theorem which is updating the beliefs about certain events given the evidence/knowledge of some other event. In this article, we discuss in the detail the answer to the question “Do we always update our beliefs?”
Say we have two events:
It is not likely to update the belief about event ‘B’ even if we know that event ‘A’ has occurred because event ‘B’ does not depend on event ‘A’, the other way of saying the same is that the events A and B are independent events.
Let’s look at an example:
We define two events in this case:
And we are interested in the value of P(A | B):
Let’s see what information/probability we can compute from the given data:
We know that there are 50 girls and 70 boys and since all of them are equally likely, P(A) would be given as follows and once we have the probability of event ‘A’, we can compute the probability of event A compliment as well:
We can also find the value of P(B | A) i.e the probability that a student is good at maths given the student is a girl and similarly we can compute the probability of the event B given A complement:
We can compute P(A | B) using the Baye’s theorem
In the above formula, we don’t have the value of P(B) but we can compute the same using the total probability theorem as the union of the event A and A complement forms the sample space
So, we get the value of P(A | B) as 5/12 which is the same as the P(A), so having knowledge about the event B does not change the belief about event A and it makes sense as being good at maths does not depend on gender and that means the event A and B are the independent events.
And P(B) is the same as P(B | A) which is the same as P(B | A complement) so knowing about A does not change the belief about B.
Two events are independent if P(A | B) = P(A)
There is a more robust way of saying that two events are independent, we know that the P(A intersection B) can be represented as below:
And if two events are independent that P(A | B) would be the same as P(A)
Let’s take some examples:
In the first example, say 3 coins are being tossed and the events A and B are defined as below
And there are two events as specified in the above image and the question we want to answer is “Are A and B independent?”
And we can be sure that these events are independent if it satisfies the below equation:
P(A intersection B) = P(A). P(B)
Below are the possible outcomes when we toss 3 coins simultaneously:
Out of these 8 possible outcomes, only 4 of them are favorable to A(event A corresponds to the case that the first coin results in Heads) and using this, we can compute the probability of A
Similarly, we can mark the outcomes which are favorable to B, and using that we can compute the probability of B and in a similar manner, we can compute the outcomes favorable to A intersection B and compute the probability of A intersection B:
We see that the probability of A intersection B is not the same as the product of P(A) and P(B) and intuitively this makes sense as well because if we know that the first toss results in heads then our belief about getting two heads would get changed and that’s why these events are not independent.
Let’s take one more example: say there are two dice being rolled and the events are defined as below:
And we are interested in knowing if the two events are independent?
We can define the set of outcomes which falls under event A and from the count of outcomes in A, we can compute the probability of A and similarly, we can note down the outcomes in event B, and in A intersection B and compute their respective probabilities
So, these two events are independent and this makes sense as well, as in set A/outcomes in the event A, we have 3 cases where the sum is 7 even if the second number is not even, there are 3 cases where the sum is 7 and the second number is even, so there are equal chances of the sum being 7 and the sum being 7 does not depend on what the second number is and hence these two events are independent.
Independence — n events:
Let’s understand the second point taking ‘n’ as 3 and we can write down the possible outcomes
The second property of independence of ’n’ events says that if we take any of the subsets, then the rule(probability of the intersection of the events must be equal to the product of the individual probabilities) for independence must be satisfied
The mutual independent condition itself ensures pairwise independence but it also ensures more than that.
Summary: In the last few articles, we covered a great number of topics, and here is a quick summary of all that: