Compound events example with tree diagram

if you flip three fair coins what is the probability that you'll get at least two tails the tree diagram below shows all the possible outcomes of flipping three coins and the top of the tree this shows us the two outcomes for the first coin and then given each of those outcomes it shows us what's possible for the second coin so if we got a heads on the first coin we could get a heads or tails on the second coin now two tails on the first coin well we could get a heads or tails on the second coin and then for each of those outcomes it shows us the different outcomes for the third coins so let's just think about what we're on this how do we represent getting at least two tails or what are the total outcomes and which of those meet our constraints of getting at least two tails so this this node right over here this is getting a heads on the third coin a heads on the second coin we just have to follow up the tree and a heads on the first coin so this is getting three heads so this is definitely not going to meet our constraint this this node right over here we have a head head this is often called a leaf if we're talking about a tree diagram a head head and a tail so that's one tail that doesn't meet our constraint of at least two tails what about this one here heads tails heads once again only one tail so that doesn't meet it heads tails tails this one does so let me color this in let me color all the ones that meet our constraints this is getting a tail on the third one a tail on the second one and a heads on the first so that's at least two tails here we have tails heads heads that doesn't meet it tails heads tails that does so let's color this one in and then tails tails well if you've got a tail on the first and the second then either of these are going to meet the constraints because you already got two tails so that one meets it that you got tails tails heads and tails tails tails both of them so let's go back to the question what is the probability that you'll get at least two tails well how many equally likely outcomes are there we're assuming this is a fair coin we see that there are 1 2 3 4 5 6 7 8 equally likely outcomes and how many of these outcomes met our constraints 1 2 3 4 4 out the eight four eight which could also be viewed is equivalent to 1/2 the probability that I'll get at least two tails is 1/2