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.