# Definition of events in independence simple coin toss

I struggle defining the experiment, the sample space and the events.

For instance, I have two ways of defining the following:

1. I have the experiment of tossing a coin twice, my sample space would be {HH, HT, TH, TT}. If I have two events, A: Getting H in the first toss, B:getting H in the second toss. Then A={HH, HT} and B={HH, TH}, where P(A)P(B) = 1/4.

2. I have the experiment of tossing a coin that is repeated twice, the sample space is {H,T}. I have two events, A:Getting H in the first toss, B:getting H in the second toss. Then A={H} and B={H}. where where P(A)P(B) = 1/4. Although I find this weird, it is how I see commonly calculated probabilities, but the experiment and events are not usually explicitly defined. (Please correct me if I am wrong).

If I want to give an example with two dependent events, then I could say the following:

1. I have the experiment of tossing a coin twice, my sample space would be {HH, HT, TH, TT}. If I have two events, A: Getting H in the first toss, B:I don't know how to define this event.

I don't want to call B:getting H in the second toss given that I got H in the first toss, because that is represented with P(B|A) and I need to specify what B is.

I also don't want to call an event like Getting two Hs because I can't identify A and B.

So, how can describe the events? and the experiment? and the sample space?

What differs between independent and dependent scenarios is the conditional probabilities. If $$B$$ is the event of getting H on the second toss, then:
• In an independent tosses scenario, $$P(B) = \sum_n P(B ,A_n ) = \sum_n P(B|A_n)P(A_n)$$ (law of total probability, where $$A_n$$ enumerates over the different outcomes of the first coin toss, H or T). $$P(B|A_n) = P(B)$$ due to independence, so $$P(B) = \sum_n P(B)P(A_n) (0.5 \times 0.5) + (0.5 \times 0.5) = 0.5$$ (assuming a fair coin)
• In a dependent tosses scenario, still $$P(B) = \sum_n P(B|A_n)P(A_n)$$, but we you have to know further information about $$P(B|A_n)$$ to be able to solve for $$P(B)$$. For example, see the "tree diagram" plot at this link.