Lesson Video: Experimental Probability | Nagwa Lesson Video: Experimental Probability | Nagwa

# Lesson Video: Experimental Probability Mathematics • First Year of Preparatory School

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In this video, we will learn how to interpret a data set by finding and evaluating the experimental probability.

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### Video Transcript

In this video, we will learn how to interpret a dataset by finding and evaluating experimental probability.

When we are calculating probability, we’re really determining the likelihood that this event will occur. There are two main ways in which we can estimate the probability of an event. So let’s consider these first. One way in which we can estimate the probability is by considering the attributes or physical properties of the event in question. For example, if we wanted to calculate the probability of rolling a five on a fair die, we would consider the number of sides on the die. The probability of one-sixth would be the theoretical probability of rolling a five.

But sometimes we can’t use theoretical probability, for example, if we knew that this die was unfair. This is an occasion where we could use experimental probability. This is when we perform an experiment, for example, throwing a die. We say that we perform repeated trials and record the outcome, that’s the result, of each trial. We can then calculate experimental probability using the formula that this is equal to the number of trials in which the outcome occurs over the total number of trials. Let’s now see how we can apply this in a practical example.

The table shows the results of a survey that asked 20 students about their favorite breakfast. What is the probability that a randomly selected student prefers eggs?

We can begin this question by having a look at the table. We can see that out of 20 students, 10 students said that eggs were their favorite breakfast, two students said that cereal was, and eight students said that toast was their favorite. When it comes to calculating the probability that a student prefers eggs, we’re really considering this as experimental probability. Although it doesn’t appear to be a traditional experiment, there has been a repeated trial of asking different students for their favorite breakfast and then a recording of the outcomes. That’s the answers that the students gave. The standard formula that we might use to calculate experimental probability can be given that this is equal to the number of trials in which the outcome occurs over the total number of trials.

However, we can rewrite this according to the context. We want to calculate the probability that a student prefers eggs. So that can be calculated as the number of students who preferred eggs over the total number of students surveyed. Then, from the table, we can see that there were a total of 10 students who preferred eggs. And we are told that 20 students were surveyed in total. Notice that even if we weren’t given the figure of 20, we could calculate this from the sum of the frequencies, that’s the number of students, of 10 plus two plus eight. So we now have that the probability of a student preferring eggs is 10 over 20. But of course we know that this fraction can be simplified further to one-half. And so we can give the answer that the probability that a randomly selected student prefers eggs is one-half.

In the next example, we’ll see how to calculate the probability when the outcomes of an experiment are given as a list.

Isabella creates a three-sided spinner using the colors red, green, and blue. She spins the spinner and records the following results: red, blue, red, green, green, green, red, red, red, green. Calculate the experimental probability of spinning green on this spinner.

We are told that Isabella has a three-sided spinner with three colors: red, green, and blue. The outcomes of the experiment to spin the spinner have been recorded. We then need to calculate the experimental probability of spinning green. The experimental probability of an event is the ratio of the number of outcomes in which this event occurs to the total number of trials in the experiment. So here we’re calculating the experimental probability of spinning green. This is equal to the number of times green was spun over the total number of spins.

Using the set of results, we can see that green was spun four times. And by counting the number of results, we can determine that the number of spins must be 10. The fraction four-tenths can be fully simplified to two-fifths. We can therefore give the answer that the experimental probability of spinning green is two-fifths. However, as probabilities can be given as fractions, decimals, or percentages, then 0.4 or 40 percent would also be valid answers.

We have now seen how we can determine experimental probability when the data is presented as a table and as a set of results. In the next example, we’ll see how we can use a bar graph to calculate experimental probability.

The graph shows the results of an experiment in which a die was rolled 26 times. Find the experimental probability of rolling a two. Give your answer as a fraction in its simplest form.

Here, we can see that we have a bar graph representing the number of times the values one to six were rolled on a die. For example, number one was rolled four times. Number two was rolled eight times. We can calculate the experimental probability of rolling a two as the number of times two was rolled over the total number of rolls. If we look at the bar chart, even if we weren’t given the figure of eight, we could read that the top of bar two comes to the value of eight. In other words, the number two was rolled eight times. We are then given that the die was rolled 26 times. However, even if we weren’t given this total number of rolls, we could calculate it by adding four, eight, eight, three, one, and two. This would give us a value of 26. The probability of rolling a two is eight over 26. But as these are both even numbers, we know that this fraction will simplify further to four over 13. And that’s our fraction in its simplest form for the experimental probability of rolling a two.

Let’s now see another example.

A game at a festival challenged people to throw a baseball through a tire. Of the first 68 participants, three people won the gold prize, 12 won the silver prize, and 15 won the bronze prize. What is the experimental probability of not winning any of the three prizes?

Although it may at first appear that there are just three different outcomes in this game, there are in fact four. It’s possible to win gold, silver, or bronze prizes or no prizes. Given the information that three people won gold, 12 won silver, and 15 won the bronze prize out of a total of 68 participants, we can calculate the number of participants who won no prizes. It will be equal to 68 minus three plus 12 plus 15. That’s 38 people. We can then use this value to find the experimental probability of not winning any prizes, since this is equal to the number of participants who won no prizes over the total number of participants. 38 people won no prizes, and there were 68 participants. This simplifies to give us the answer that the experimental probability of not winning any of the three prizes is 19 over 34.

In the final question, we will use a given experimental probability and a value for the number of outcomes to calculate the total number of trials in an experiment.

The experimental probability that a coin lands on tails is three-sevenths. If the coin landed on tails 30 times, how many times was it tossed in the experiment?

In this question, we are given the experimental probability that a coin lands on tails. This probability has been calculated using the data from an experiment with repeated trials of tossing a coin. We are also given that the number of outcomes of landing on tails, that’s the number of times the coin landed on tails, is 30. We can recall that, in general, experimental probability is calculated as the number of trials in which the outcome occurs over the total number of trials. In the context of this problem, we would say that the experimental probability of a coin landing on tails is equal to the number of times the coin landed on tails over the total number of times the coin was tossed.

Usually, when we apply this formula, we’re doing it to calculate the experimental probability when we are given the two values on the right-hand side. However, here we know the experimental probability, we know the number of times the coin landed on tails, and we want to calculate the total number of times the coin was tossed. We can then substitute in the values of three-sevenths for the experimental probability and 30 for the number of times the coin landed on tails. We can then solve this by cross multiplying. Three times the total number of times the coin was tossed is equal to 30 times seven, and 30 times seven is equal to 210. We can then divide both sides by three. 210 divided by three is equal to 70. The answer is that the coin was tossed 70 times in this experiment.

We can now summarize the key points of this video. Firstly, we saw that experiments can be used to estimate the probability of an event occurring. To gather data to calculate experimental probability, we perform repeated trials and record the outcome of each trial. It’s worth noting that the more trials we perform, the more accurate an estimate of results we will get. The experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment. We can remember this in the general formula that experimental probability of an event is equal to the number of trials in which the outcome occurs over the total number of trials. And finally, as we saw across the range of our examples, we can calculate experimental probability from data presented in the form of a statement, a set of results, a table, or a graph.

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