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Alright, so we are going to toss a coin two times. And we're going to assume it's a fair
coin for this. Does knowing that a head occurs on the first toss change the probability of
getting a heads on the second toss? Well, this is actually asking about the independence
of the events, of getting a heads on the second toss. What's the probability of getting a
heads on the second toss? And what's the probability of getting a heads on the second toss, given
the first toss was a heads itself.
So we're going to take a look at this and see if we can prove these events are independent
or not. So when I'm tossing coins two times, there's four possible combinations that can
occur. I can get a heads and a heads, so a heads on the first toss, a heads on the second
toss; a heads and then tails; a tail then a head; or a tail and a tail.
So if we look at just the unconditional probability that the second toss is a head, I see that
there are two different ways this can occur out of the four possible. Since we've got
a fair coin, we can assume all of these are equal likely. So I just need to count the
number of ways that this event occurs out of the number possible.
Now the conditional one says that, okay, I want to find the probability that the second
is a head, given that the first toss was heads. So whenever we have a conditional probability,
that tells me I have a new universe or a new sample space that I'm dealing with. I can
only look at these outcomes. The first toss was a heads, so that was a heads and a heads,
or a heads and a tails. Those were the only two outcomes under this given information.
Under this sample space, find the probability that the second toss is a head.
Well, when I look at this new sample space, again, we have that fair coin, so these are
equal likely. It only occurs one out of the two times. So I'm using my counting property
for finding the probability. And what I see is that these probabilities are the same.
And so that tells me that the event that the second toss is a head, is independent of the
event that the first toss was a head, as our given piece of information.
So remember when we're proving that two events are independent of each other, we have this
general statement that must be true for independence; the probability of A given B has to equal
the probability of A if, and only if, A and B are independent.
If the events happen to be dependent, what would happen is that when you look at this
statement, you'll find it'll be a false-- that when you compute the probability of A
given B, if it's not the same as probability of A, then we're guaranteed that they are
not independent of each other. So if this is true, that's only if A and B are not independent.