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# Example: All the ways you can flip a coin

Manually going through the combinatorics to determine the probability of an event occuring. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

Find the probability
of getting exactly two heads when flipping three coins. So let's think about
the sample space. Let's think about all of
the possible outcomes. So I could get all heads. So on flip one I get a
head, flip two I get a head, flip three I get a head. I could get two heads
and then a tail. I could get heads,
tails, heads, or I could get heads, tails, tails. I could get tails, heads, heads. I could get tails, heads, tails. I could get tails, tails, heads. Or I could get tails,
tails, and tails. These are all of
the different ways that I could flip three coins. And you can maybe
say that this is the first flip, the second
flip, and the third flip. Now, so this right over
here is the sample space. There's eight possible outcomes. Let me write this, the
probability of exactly two heads, I'll say H's
there for short. The probability of
exactly two heads, well what is the size of
our sample space? I have eight possible outcomes. So eight, this is
possible outcomes, or the size of our sample
space, possible outcomes. And how many of those
possible outcomes are associated with this event? You could call this
a compound event, because there's more
than one outcome that's associated with this. Let's think about
exactly two heads. This is three heads, so
it's not exactly two heads. This is exactly two
heads right over here. This is exactly two
heads right over here. There's only one head. This is exactly two heads. This is only one head,
only one head, no heads. So you have one, two, three
of the possible outcomes are associated with this event. So you have three
possible outcomes. Three outcomes
associated with event. Three outcomes
satisfy this event, are associated with this event. So the probability of
getting exactly two heads when flipping three
coins is three outcomes satisfying this event over
eight possible outcomes. So it is 3/8.