Main content

## Compound probability of independent events using diagrams

# Compound events example with tree diagram

## Video transcript

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. At 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 coins. If we got a heads
on the first coin, we could get a heads or a
tails on the second coin. Got a tails on the
first coin, well, we could get a heads or a
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 where 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 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 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 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've 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 of the 8. 4/8, which could also be
viewed is equivalent to 1/2. The probability that I'll get
at least two tails is 1/2.