Video Transcript
In this video, we will learn how to
find and interpret the probability of compound events. We’ll consider both independent and
dependent compound events.
When we consider the probability of
a simple event, that’s the probability of a single event that is not dependent on
any other event. For example, a coin toss is a
simple event. It’s a single event that results in
one of two outcomes, heads or tails.
We remember that the probability of
event A is all the ways event A can happen over all of the possible outcomes. For a simple event, such as a coin
flip, we find the probability of getting heads is one over two. There’s one way the coin will be
heads out of the two possible options. But if we flip the coin twice, it
no longer meets the definition for a simple event, as it’s not a single event. By tossing the coin twice, we’ve
moved from a simple event to a compound event.
And now, we want to know how would
we go about finding the probability of a compound event. The probability of a compound event
is still equal to the number of successful outcomes over all possible outcomes. But it takes a few additional steps
to calculate it. Let’s look at an example of finding
the probability in a compound event.
If we throw a coin twice, what is
the probability of getting heads both times?
We remember that probability is the
number of successful outcomes over all possible outcomes. The coin is tossed twice. We can use a tree diagram to show
all possible outcomes. In the first toss, you have a
possible outcome of heads or tails. The probability of getting heads on
the first toss is one-half. And the probability of getting
tails on the first toss is one-half.
And now, we need to consider two
different situations. We consider the second toss if the
first toss was heads. While the second toss can still
only be heads or tails, the probability that it is heads is one-half and the
probability that it’s tails is again one-half. And now, we consider if the first
toss was tails, the second toss could be heads or tails. And each of those options has a
probability of one-half.
Using the tree diagram, we can see
all possible outcomes, heads, heads; heads, tails; tails, heads; or tails,
tails. We want to know the probability of
getting heads both times. And that happens once out of the
four possible outcomes. We can see that the probability of
getting heads the first time was one-half and the probability of getting heads the
second time was one-half.
What’s happening here is that to
find the probability we get heads both times, we take the probability that we got
heads the first time and then we multiplied it by the probability that we got heads
the second time. One-fourth is equal to one-half
times one-half. We remember that we can also write
probability as a decimal. So, the probability if we threw a
coin twice of getting heads both times is one-fourth, or 0.25.
Here’s another example of simple
compound event probability.
If these two spinners are spun,
what is the probability that the sum of the numbers the arrows land on is a multiple
of five?
To find this probability, we’ll
need to consider spinning the first spinner event one and spinning the second
spinner event two, making this a compound event. If we want to know the probability
that the sum of the spinners is a multiple of five, we need the multiple of five
outcomes over the total outcomes. There are a few methods we could
use. We could create a tree diagram for
spinner one and spinner two. However, because we need to sum the
results of spinner one and spinner two, a table is probably a better choice.
In the first row and the first
column, we would add if spinner one landed on one and spinner two landed on two,
this sum would be three. And then, we’ll consider as spinner
one landed on one and spinner two landed on four. This sum would be five. We fill in the rest of our table
with the correct sums. In the table, we have all possible
outcomes. And we can circle the ones that are
multiples of five.
We’ve identified five of the
results that are multiples of five out of the 21 possibilities. The probability that the sum of the
spinners is a multiple of five is five over 21. The fraction can’t be reduced any
further, so five over 21 is the final answer.
Before we move on and look at other
examples, we need to look a little bit more closely at compound events. There are two types of compound
events. We have compound events that are
independent from one another and compound events that are dependent on one
another. So far, in our previous examples,
we’ve only been considering independent events.
Tossing a coin multiple times is
independent. When we flip a coin, what happens
the first time does not affect what happens the second time. Whether we threw heads or tails on
the first throw makes no difference to the probability of what will happen on the
second throw. The same thing is true for
spinners. The first spinner — or in fact, the
first spin — has no bearing on what the next ones will be.
On the other hand, if we have a bag
full of marbles and we find the probability of removing a yellow ball and we do not
replace it, we are changing the number of outcomes for the second event. If we were looking for the
probability of yellow first and green second, we first find the probability of
selecting a yellow ball. And then, we need to calculate the
probability that we would select a green given that the yellow has already been
removed.
If we have two events A and B, if
the fact of A occurring does not affect the probability of B occurring, the events
are independent. And in that case, we say the
probability of A and B occurring is the probability of A times the probability of
B. The compound events are dependent
if the fact of A occurring does affect the probability of B occurring. And in that case, the probability
of A and B is equal to the probability of A times the probability of B given A.
Let’s consider how we find the
probability of two events happening if we already know that they are independent
events.
A and B are independent events,
where the probability of A is one-third and the probability of B is two-fifths. What is the probability that the
two events A and B both occur?
We know that these are independent
events, which means that event A occurring does not affect the probability of B. And we can say that the probability
of A and B both occurring is the probability of A times the probability of B. The probability of A and B is equal
to one-third times two-fifths. We multiply the numerators and then
multiply the denominators to get two fifteenths. And we can’t simplify that any
further. So, that is the final answer. The probability that the two events
A and B both occur are two fifteenths.
This question was really
straightforward because we were told that the events were independent and we were
given both of their probabilities. It won’t always be this
straightforward. And in many cases, we’ll need to
determine whether or not the events are, in fact, independent. This is one of the times when we’ll
have to decide if these events are independent or dependent.
A bag contains eight red balls,
seven green balls, 12 blue balls, 15 orange balls, and seven yellow balls. If two balls are drawn
consecutively without replacement, what is the probability that the first ball is
red and the second ball is blue?
We should notice that we are
considering two different events, which means these are compound events. And as we’re drawing balls out of
the bag, we are not replacing them. This means the first thing that
happens will affect the probability of the second outcome. And that tells us that these are
compound dependent events. When we’re looking for the
probability of compound dependent events, it will be equal to the probability of
event A times the probability of event B given that event A does occur.
In this case, we want the
probability of drawing red and then blue. We know probability is equal to the
successful outcomes over all possible outcomes. And we first need to find the
probability that we choose red on the first draw. When we began, there are eight red
balls. At the beginning before we draw,
there are a total of seven plus seven plus eight plus 12 plus 15 balls in the bag,
for a total of 49. On the first draw, the probability
of drawing red is equal to eight over 49.
Now, if we drew a red on the first
draw, there would be 48 remaining balls. And out of those 48 balls, 12 of
them are blue. The probability of drawing red and
then blue would be eight over 49 times 12 over 48. We can reduce these fractions
before we multiply. 12 over 48 simplifies to
one-fourth. And eight over four simplifies to
two over one. So, the probability of selecting
red and then blue is equal to two over 49, which can’t be simplified any
further. So, it’s the final answer, two
forty-ninths.
We’ll now look at another example
where we’re not told if the events are independent or dependent.
A meteorite lands at random in a
field containing a lot of sheep. Considering the size of the
meteorite, the size of the field, and the amount of space taken up by the sheep, the
probability that some of the sheep are harmed in the incident is one out of 35. Nearby, a panel falls off a
helicopter and into a field of cows. The panel is quite large and the
field is fairly full of cows. So, the probability of some cows
being harmed is one-third. What is the probability that no
animals were injured in the two incidents?
We’re interested in
probability. And there were two incidents. So, we know that the probability
will be dealing with compound events. If we consider the two events, a
meteorite strike and a helicopter panel strike, are these two events independent or
dependent? Does the fact of the first event
affect the probability of the second event? If a sheep is harmed, does that
change the likelihood that a cow will be harmed?
Since the fact of the first event
does not affect the probability of the second event, we can say that these compound
events are independent. Since we’re dealing with compound
independent events, the probability that event A and B both occur is the probability
of A times the probability of B.
We’re interested in the probability
that the sheep are safe and the cows are safe. We have to be careful here because
we were only given the probability that a sheep is injured. To find the probability that the
sheep are safe, we can take the probability that the sheep are injured and subtract
it from one. If there is a one out of 35 chance
that the sheep are injured, there’s a 34 out of 35 chance that they are safe. And so, we can take that
information in and plug it into our formula. The probability that the sheep are
safe is 34 out of 35.
We need to follow the same process
for the cows. The probability that the cows are
safe is one minus one-third, which equals two-thirds. And we can plug that in for event
B. The probability that the sheep and
the cows are all safe will be equal to the probability that the sheep are safe times
the probability that the cows are safe. 34 over 35 times two-thirds and
then we get 68 over 105. The probability that no animals
were injured in the two incidents is 68 out of 105. This fraction cannot be reduced, so
it’s in its final form.
At this point, we should note
something. Probabilities with compound events
are probabilities that are dealing with more than one simple event. And that means we can have more
than two events we’re considering. You could have a coin flipped three
times or 15 times. This would be an independent
event. And so, to find the probability of
independent events where they’re more than two events, we follow the same
procedure. We would multiply the probability
of event A by the probability of event B by the probability of event C.
Dependent events would follow the
same procedure, but we would need to be a bit more careful. To find the probability of three
dependent events, A and B and C, we would find the probability of A times the
probability of B given A. And then, we would have to multiply
that by the probability of C given both A and B.
Let’s summarize what we’ve
learned. Compound probability is concerned
with the probability of more than one event occurring. Compound events may be independent
or dependent. For independent compound events,
the probability of A and B is equal to the probability of A times the probability of
B. To calculate the probability of
dependent compound events, we say the probability of A and B is equal to the
probability of A times the probability of B given A.
