Probability- independent events

[C.E]

1. A fair coin is tossed n times, let E be the event that the first toss is a head and Fk be the event that there are a total of k heads. For which values of n, k are E and fk independent events?

3. I don't see how these events can possibly be independent (surely the first influences the second). Could somebody please explain how to do this question?

 

[Defennder]

You can approach this question mathematically instead of by conceptual reasoning. E and F_k are independent if P(E|F_k)=P(E) and P(F_k|E)=P(F_k).

Now you just have to find expressions for the above. Use Bayes theorem and the definition of conditional probability. After solving the above, you should be able to find n in terms of k.

 

[Architeuthis Dux]

I understand your confusion about the concept of independent events. In probability, two events are considered independent if the occurrence of one event does not affect the probability of the other event. In the given scenario, the event E (first toss is a head) and Fk (total of k heads) can be independent for certain values of n and k, as long as the first toss does not have any influence on the number of heads in the total outcome.

To understand this better, let's take the example of flipping a fair coin 3 times (n = 3). In this case, E would be the event of getting a head on the first toss and Fk would represent the event of getting a total of k heads (k = 0, 1, 2, 3).
We can see that for k = 0, 1, 2, 3, the probability of Fk does not change regardless of the outcome of the first toss (E). Therefore, in this case, E and Fk are independent events.

However, if we consider a different scenario where n = 2 (coin is tossed only twice), then E and Fk are not independent events. This is because the outcome of the first toss (E) will directly affect the number of heads in the total outcome (Fk).

In summary, the independence of events E and Fk depends on the values of n and k. For certain values, they can be independent while for others, they are dependent. I hope this explanation helps you understand the concept better.