In this video, we will learn how to use probability to predict the actions of a larger group using a sample to find the expected value. We can start by reminding ourselves that probability refers to the likelihood or chance of an event occurring. We can find the probability of an event in two ways.
The first way is by using the theoretical probability, which is when we use mathematical reasoning about possible outcomes. For example, if we were to flip a fair coin with heads on one side and tails on the other, then we can say that the probability of getting tails would be equal to one-half since there’s one tail out of two possible outcomes. The second way to find the probability of an event is by using the experimental probability or relative frequency. This is when we use the outcomes of a repeated experiment to find the probability.
We can use either the theoretical probability or the experimental probability to find the expected value. This is the value that we would expect to happen on average over a large number of trials. We can calculate the expected value by multiplying the probability of an event occurring by the number of trials or experiments that are performed. Let’s have a look at some examples of how we can calculate the expected value of an event.
The probability that a biased die will land on an even number is 0.6. If the die is rolled 80 times, how many times is it expected to land on an even number?
So here we have our die. We’re told that it’s biased, which means that every number is not equally likely. We’re told that the probability of getting an even number is 0.6. And we need to calculate how many even numbers would be expected if the die is rolled 80 times.
To find this expected value, we multiply the probability of the event occurring by the number of trials. So to find the expected value of even numbers, we take the probability of getting an even number and multiply it by 80 since it was rolled 80 times. Therefore, we have 0.6 times 80, which is 48. So we would expect there to be 48 even numbers when the die is rolled 80 times.
Jacob plays a card game with his friend, and the result can be either win, lose, or draw. Each time they play, the probability that Jacob will win is 0.5 and the probability he will lose is 0.3. If they play 50 games, what is the expected number that will end in a draw?
We’re told that in this card game there are only three possible outcomes: win, lose, or draw. We’re told that the probability that Jacob wins is 0.5 and the probability that he loses is 0.3. In order to work out the probability then that he draws, since these three events are the only possible outcomes, then they will add up to one. Meaning that the probability that Jacob draws is 0.2.
We now need to calculate that if 50 games are played, how many of these will end in a draw. This means that we work out the expected value by multiplying the probability of an event by the number of trials. The probability of the event occurring here is the probability of getting a draw. We can therefore fill in our values that the probability of getting a draw is 0.2 multiplied by the number of trials, which is 50 games. As 0.2 is equivalent to one-fifth, then a fifth of 50 is equal to 10. So our answer is that we would expect 10 games to end in a draw.
In the following examples, we’ll see how we can use the experimental probability of an event to work out the expected value.
In a survey of 400 tourists who visited Egypt, 160 were from Arab countries, 120 were from Europe, 40 from Latin America, and 80 from Australia. If the total number of tourists who visited Egypt in a month was 5000, how many of them are expected to be from Europe?
Let’s start by picking out the key information. We’re asked about the tourists that are from Europe. From the survey results, we can see that 120 were from Europe and that there are 400 tourists in total. So we have a fraction of 120 out of 400. Notice that even if we hadn’t been given the value of 400, we could’ve calculated this from 160 plus 120 plus 40 plus 80. In the survey results then, we have 120 tourists out of 400 were from Europe.
We now need to expand this to work out of the 5000 tourists in a month how many of those would be from Europe. To find this, we calculate the expected value. This is calculated by multiplying the probability of an event occurring by the number of trials. So therefore, to find the expected value of tourists from Europe, we multiply the probability that a tourist is from Europe, which we find in the experimental probability from the survey, and multiply it by the number of tourists, which gives us 120 over 400 multiplied by 5000.
We can reduce the fraction 120 over 400 to three-tenths. And then we have a 10 on the denominator and 5000 on the numerator. So we can cancel down to three times 500, giving us 1500. So we would expect that out of 5000 tourists that 1500 would be from Europe.
The table shows the results of rolling a die 78 times. Using this information, how many times is the number two expected to appear if the die is rolled 234 times?
So here we have a table showing the results of this experiment of rolling the die. We can see that the total number of rolls does indeed add up to 78. We’re interested in the number of times that the number two occurs. This will be 17 rolls out of 78. We could therefore write that the probability of rolling a two from this experiment is 17 over 78.
Notice that this is the experimental probability or relative frequency and not the theoretical probability of rolling a two, which would usually be one-sixth. We’re going to take this experimental probability and extend it to find the result for 234 rolls of a die. To do this, we use the expected value, which is equal to the probability of an event multiplied by the number of trials.
So here the probability of our event is 17 over 78 multiplied by the number of trials, which is 234. We can simplify this calculation by noticing that 78 divides into 234 three times, and so 17 times three is 51. And so we would expect two on our die to appear 51 times out of 234 rolls.
A factory produces two types of shirts, A and B. A sample of 100 shirts from each of five shopping centers was observed to see how many of each type were sold. The results are shown in the table. If the factory sells 3000 shirts, how many of them do you expect to be of type A?
So let’s have a look at the data in the table. We can see that each shopping center from one to five sells 100 shirts. And the sales of type A and B are listed for each shopping center. If we added up the sales of type A across all shopping centers, we would find that that is equal to 227 shirts. And the sales of all of type B will give us 273. This means that if we wanted to select a shirt of type A from these five shopping centers, then the probability of picking a shirt of type A would be equal to 227 over the total number of shirts, which would be 500, since we know that’s the sum of the total shirts of type A and B. Or equally it’s the five lots of 100 shirts in each shopping center.
We now need to calculate out of 3000 shirts how many would we expect to be of type A. We can recall that the expected value is equal to the probability of an event multiplied by the number of trials. So here our probability will be 227 over 500. And the number of trials will be the number of shirts we’re looking at, which is 3000. We can then simplify this calculation to 227 multiplied by six, which is equal to 1362 shirts of type A.
And now let’s summarize what we’ve learnt in this video. The expected value is what we expect to happen on average over many trials or experiments. It is calculated by multiplying the probability of an event occurring by the number of experiments or trials. And finally, we can use either the theoretical or experimental probability for the probability of an event occurring.