Video Transcript
If all of the possible outcomes
from an experiment are equally likely to occur, then calculating the probability of
an event becomes a simple matter of counting up the number of outcomes that make up
the event, and express that as a fraction of the total number of possible
outcomes.
So, for example, if we roll a fair
six-sided dice, there are six possible outcomes. We can get a one, two, three,
four, five, or six and they’re all equally likely. Now one of the outcomes is a
three. So there’s one way to get a
three. So the probability of getting a
three is this one divided by the total number of outcomes, six. So the probability of getting a
three is one over six.
So when all of the outcomes are
equally likely, to work out the probability of an event, we get the number of ways
to get the result we’re looking for and divide that by the total number of outcomes
that we have altogether. But remember, our method of
counting outcomes to work out probabilities, only works when the outcomes are all
equally likely. So let’s look at an example where
that isn’t the case.
If we buy a lottery ticket, there
are two possible outcomes. We can win, or we can lose. But most lotteries, those two don’t
have the same probability. We’re much more likely to lose than
we are to win. So they’re not equally likely
outcomes. So although we have two outcomes in
total, and there’s one way to win. The probability of winning is not
just one divided by two. It’s not a half because those two
outcomes were not equally likely. Remember, this was much more likely
to occur than this. So we can’t just do the
counting. So, as always, something to work,
to look out for, are the outcomes equally likely?
Okay. Now let’s go back to our dice
example, where the outcomes were all equally likely. Let’s work out the probability of
the event of getting a factor of twelve, when we roll our dice. So the first thing we need to do,
is see which of the results would go into event 𝐸, so which of those numbers are
factors of twelve. Well one is, two is, three is, four
is, five isn’t, and six is. So that’s five equally likely
outcomes making up event 𝐸. So the probability that event 𝐸
occurs, that we get a factor of twelve when we roll our dice, well there’re five
ways of that happening, so that’s five, out of the six possible outcomes that we had
in total, so that’s five over six. So with equally likely outcomes,
it’s just a matter of counting up the cases. There are five ways of getting a
factor of twelve out of six possible equally likely ways of things turning out in
the end.
Now this works in slightly more
complex situations too. So let’s say we roll two fair dice
and add the scores together. We can put the results in a table
like this. We’ve got thirty-six possible
equally likely outcomes. So, for example, we could get a one
on the first dice and a one on the second dice, making a total of two. We could get a one on the first
dice and a two on the second dice, making a total of three, and so on. So let’s ask the question, what is
the probability of getting a result of nine? Well there are thirty-six possible
equally likely outcomes and one, two, three, four of them result in a nine. So the probability of getting a
nine is four out of thirty-six, four ways of getting a nine out of thirty-six
possible equally likely outcomes in total.
Now let’s ask a question, what’s
the probability of getting an odd result? Well we can see that eighteen out
of the thirty-six equally likely outcomes give odd results, odd-numbered
results. So the probability of getting an
odd number is eighteen over thirty-six. Now we could simplify that to a
half, but we don’t have to. In probability, quite happy to
leave that eighteen over thirty-six. Again, it’s kind of more
informative than a half in some ways because there are eighteen ways of getting what
we’re looking for, out of the thirty-six total possible ways that there are of
getting results.
Okay. Now let’s consider an experiment
where we roll two fair dice, and we multiply their scores together. Again, we’ve got thirty-six
possible equally likely outcomes. We could get a one on the first
dice and a one on the second dice which would — one times one gives us a result of
one. Or, we could get a one on the first
dice and a two on the second dice, and one times two gives us a result of two, and
so on, and so on, for all thirty-six different examples. So when we write out all the
results in a sample space, in a table, each cell is equally likely to occur. And again, we can just count up the
ways of results happening.
So let’s ask the same questions
again, what’s the probability of getting a result of nine? Well there are thirty-six possible
outcomes, all equally likely. But only one of them here, if we
get a three and a three, generates a result of nine. So there’s one way out of
thirty-six of getting a nine.
And, so what’s the probability of
getting an odd-numbered result? Well now that we’re multiplying the
numbers together on the dice, there’s only nine ways of getting an odd result, so
nine out of thirty-six. The probability of getting an odd
number result in this situation is a lot lower than it was last time.
So to sum all that up, if all of
our outcomes are equally likely in an experiment, if we’ve got a sample space of 𝑆
and 𝐸 is an event that we’re looking for, to work out the probability of event 𝐸
occurring, we just count up how many ways are there of getting 𝐸 versus how many
ways were there in the-in the sample space altogether. And then just express that as a
fraction.
