Lesson Video: Probability of Simple Events Mathematics

In this video, we will learn how to find the probability of a simple event.

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Video Transcript

In this video, we’ll learn how to find the probability of a simple event. We’ll begin by defining what probability is. And then we’ll see how we can move from using words like “likely” or “certain” to describe the probability of an event to using a number to describe its probability.

Probability is the likelihood or chance of an event occurring. We often see probabilities represented as a scale. And we use words like “impossible” and “certain” to describe these probabilities. The number that we used to represent an impossible event is zero. And for a certain event, it’s the number one. Events which are equally likely as unlikely can be described in numbers as a fraction, decimal, percentage as one-half, 0.5, or 50 percent. We can describe any probability using a fraction, decimal, or percentage.

In order to work out the probability of an event occurring, we work out the number of favorable outcomes and divide it by the total number of outcomes. Let’s take the example of rolling a die that has the numbers one to six on it. Let’s say we wanted to calculate the probability of rolling a two. We could write this as 𝑃 brackets two. The number of favorable outcomes would be the number of twos on the die and the total number of outcomes would be six, since there’s six different values on the die. And so the probability of rolling a two on this die is one-sixth.

We can also define the probability of an event a little more formally. If 𝐴 is an event in a sample space 𝑆, then the probability of event 𝐴 occurring is the probability of 𝐴 is equal to 𝑛 brackets 𝐴 over 𝑛 brackets 𝑆, where 𝑛 brackets 𝐴 represents the number of elements in event 𝐴 and 𝑛 brackets 𝑆 represents the number of elements in the sample space 𝑆. This might all sound a lot more complicated, but let’s return to the example of rolling the die. When we consider the probability of rolling a two, the value on the numerator is the number of elements in event 𝐴. So in the case of the die, that’s how many values there are that are the number two. Then on the denominator, it’s the number of elements that there are on the die in total. And that’s six, since there are six different numbers. And so, the probability of rolling a two is one-sixth.

Before we look at some examples, there’s one very important thing to remember about probability. And we’ve already seen this when we looked at the number line. The probability of an event must be between zero and one. For an event 𝐴, we can say that zero is less than or equal to the probability of 𝐴 is less than or equal to one. In other words, the probability of an event must be between zero and one, although of course it may be exactly equal to zero or exactly equal to one. Let’s now take a look at some examples. And in the first example, we’ll find the probability of an event where there are three possible outcomes.

A bag contains seven white marbles, eight black marbles, and seven red marbles. If a marble is chosen at random from the bag, what is the probability that it is white?

We can recall that to find the probability of an event occurring, we take the number of favorable outcomes and divide it by the total number of outcomes. Here, we’re asked to calculate the probability of getting a white marble. And we can write this as 𝑃 brackets 𝑊 for white. We then need to establish how many white marbles there are. And we’re told that there are seven. So this will be the value on the numerator. The value on the denominator will be the total number of marbles. So we’re told that there are seven white, eight black, and seven red. So we add these values together. This gives us the fraction seven over 22. We can’t simplify this fraction any further. So the probability of getting a white marble is seven over 22.

We’ll now look at another example of calculating the probability of an event.

What is the probability of selecting at random a prime number from the numbers eight, nine, 20, 19, three, and 15?

In order to answer this question, we’ll need to recall two things, firstly, how we find the probability of an event, and, secondly, what a prime number is. In order to calculate the probability, then we can use the formula to calculate the probability of an event 𝐴 as the number of elements in event 𝐴 divided by the number of elements in a sample space 𝑆. We may also see this formula written as the probability of an event is equal to the number of favorable outcomes over the total number of outcomes. Here, we’re trying to calculate the probability of selecting a prime number. A prime number is a number that has exactly two factors, one and itself.

So let’s consider the list of numbers that we were given. The first three values eight, nine, and 20 are not prime because they have more than two factors. Next, 19 and three are both prime numbers. However, 15 is not a prime number. When it comes to using the formula then, the number of prime numbers that we have is two. That’s three and 19. The value on the denominator will be six, since there were six numbers in total. It’s always good to simplify our fractions where we can. And so we can give the answer that the probability of selecting a prime number from the given list of numbers is one-third.

In the next example, we’ll see how we can find the number of elements in the sample space, given the number of elements in the event and the probability of a different event.

A bag contains 24 white balls and an unknown number of red balls. The probability of choosing at random a red ball is seven over 31. How many balls are there in the bag?

Because we’re calculating a probability, we can recall that to find the probability of an event 𝐴, we calculate 𝑛 brackets 𝐴 divided by 𝑛 brackets 𝑆, where 𝑛 brackets 𝐴 is the number of elements in event 𝐴 and 𝑛 brackets 𝑆 is the number of elements in sample space 𝑆. Let’s have a closer look at this question then. We’re told that there are 24 white balls in the bag, but an unknown number of red balls. We can use the same style of notation then to note that the number of white balls must be 24. We’re not told the number of red balls, but we can use a variable such as 𝑥 to represent this. The total number of balls in the bag can be written as the number of white balls, that’s 24, plus the number of red balls, which we’ve defined as 𝑥. The total number of balls in the bag will be the number of elements in the sample space. So we can say that 𝑛 brackets 𝑆 is equal to 24 plus 𝑥.

Next, we can have a look at the probabilities. And we’re told that the probability of picking a red ball is seven over 31. We can now write this probability formula in terms of finding the probability of a red ball. Then we’ll see if we have enough information to solve to find the value of 𝑥. To calculate the probability of getting a red ball, well, the number of elements in any event 𝐴 for a red ball would correspond to the number of red balls in the bag. The denominator, which is the number of elements in sample space 𝑆 in this context, is simply the total number of balls in the bag. We can then plug in the values that we have for these expressions. So we have seven over 31 is equal to 𝑥 over 24 plus 𝑥.

Now, we can take the cross product and solve this for 𝑥. We then distribute seven across the parentheses, which gives us 168 plus seven 𝑥 is equal to 31𝑥. Then we can subtract seven 𝑥 from both sides. Finally, dividing both sides by 24, we find that seven is equal to 𝑥, and so 𝑥 is equal to seven. Remember that we defined the number of red balls to be 𝑥, and so we’ve worked out that the number of red balls must be equal to seven. It can be very tempting to stop here. But remember, we’re asked how many balls there are in total in the bag. We’ve already worked out an expression for the number of balls in the bag. It was that given by this number of elements in the sample space. So the total number of balls is 24 plus 𝑥, which was seven. And when we add those together, we get the value of 31. Therefore, we can give the answer that there must be 31 balls in the bag.

In the final example, we’ll see how we can find the total number of elements in a sample space given the probabilities of two events and the number of occurrences of a third event.

A bag contains an unknown number of marbles. There are three red marbles, some white marbles, and some black marbles. The probability of getting a white marble is one-third and the probability of getting a black marble is one-half. Calculate the number of marbles in the bag.

Because we have a probability question here, we can recall that to find the probability of an event 𝐴, we calculate the number of elements in event 𝐴 which we can write as 𝑛 brackets 𝐴 and we divide by the number of elements in the sample space 𝑆 which we can write as 𝑛 brackets 𝑆. If we have a look at the question, we’re told that there are three different colors of marbles in the bag: red, white, and black. So let’s jot down some information from the question. We can use the same notation that the number of red marbles must be 𝑛 brackets 𝑅, which we could say is equal to three.

We aren’t given the number of white marbles or black marbles, but let’s define the number of white marbles as 𝑤 and the number of black marbles as 𝑏. Knowing these three values or expressions for these values will allow us to write down that the total number of marbles in the bag, which is the same as the number of elements in the sample space, the number of marbles in the bag. That’s three plus 𝑤 plus 𝑏. Next, let’s look at the probabilities. We’re told that the probability of getting a white marble is one-third and the probability of getting a black marble is one-half.

Now, if we look at the information that we’ve jotted down, we can see that we have two unknown quantities, 𝑤 and 𝑏, which is the number of white marbles and the number of black marbles. What we’ll aim to do is to get enough information to define two equations to solve for these two unknowns. We can write the general probability formula in terms of finding the probability of a white marble. In this case, it will be equal to the number of white marbles over the total number of marbles.

Since we were given that the probability of getting a white marble is one-third, we have that on the left-hand side and then, on the right-hand side, the number of white marbles was equal to 𝑤. And the total number of marbles is equal to three plus 𝑤 plus 𝑏. Taking the cross product, we have three plus 𝑤 plus 𝑏 is equal to three 𝑤. We can then subtract the 𝑤 from both sides to give us that three plus 𝑏 is equal to two 𝑤. We can’t do anything more with this equation at the minute, but let’s label it equation one. And we’ll move it off to one side to clear some space. And hopefully, we can get a second equation in terms of 𝑤 and 𝑏, which we can then solve simultaneously to find 𝑏 and 𝑤.

What we can do next, then, is to write the probability formula in terms of finding the probability of getting a black marble. This time, we’ll be calculating the number of black marbles over the total number of marbles. And so, on the left-hand side, we’ll have one-half since that’s the probability of getting a black marble. And on the right-hand side, we’ll have 𝑏 divided by three plus 𝑤 plus 𝑏. We can take the cross product to give us three plus 𝑤 plus 𝑏 is equal to two 𝑏. Subtracting 𝑏 from both sides gives us three plus 𝑤 is equal to 𝑏. And so we have found a second equation in terms of 𝑤 and 𝑏.

If we look at these two equations, there are a number of different ways in which we could solve these. But if we look at the second equation, we have an expression for 𝑏. So we can substitute this into equation one in place of 𝑏. This gives us three plus three plus 𝑤 is equal to two 𝑤. We can then simplify this and subtract 𝑤 from both sides, which gives us that six is equal to 𝑤. This means that the number of white marbles is six.

To find 𝑏, the number of black marbles, we can substitute 𝑤 equals six into either equation, but let’s plug it into equation two. This gives us three plus six is equal to 𝑏. And therefore nine is equal to 𝑏, so 𝑏 is equal to nine. The number of black marbles is nine. To find the total number of marbles in the bag, we already have an expression for it. It’s three plus 𝑤 plus 𝑏. So when we plug in the values 𝑤 equals six and 𝑏 equals nine, we have three plus six plus nine. Working this out, we have the answer that the number of marbles in the bag must be 18.

And, of course, it’s always worthwhile checking our answers. We were given that there are three red marbles. We worked out that there must be six white marbles and nine black marbles. And those do indeed add up to 18. In terms of the probabilities, we were told that the probability of white is one-third and the probability of black is one-half. We could work out the probability of getting a red as three out of 18, which simplifies to one-sixth. When we add the three probabilities of one-sixth, one-third, and one-half, we get a total of one. Since the sum of the probabilities must be one, then this will confirm that the values that we worked out to get a total number of marbles as 18 must be correct.

We can now summarize the key points of this video. We saw that there are two formulas that we can use to calculate the probability of an event. The second formula is particularly useful when we’re dealing with the probabilities of a number of events, as we saw in a question like the previous one. And finally, we saw that all probabilities must lie in the interval zero, one. A probability that’s closer to zero would indicate an event that is less likely, and a probability that’s closer to one would indicate an event which is more likely.

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