Lesson Video: Theoretical Probability Mathematics

Video Transcript

In this video, we will learn how to interpret a data set by finding and evaluating theoretical probabilities. We begin by recalling that the probability of an event is the likelihood of it happening. And as such, the higher the probability of an event, the more likely it is to occur. All probabilities have values in the range from zero to one. For example, an event with a probability of zero can never occur and is therefore impossible, whereas an event with probability equal to one is certain to happen.

As we can see from the number line, we can express probabilities as both fractions and decimals, though in many examples we’ll be asked to express the probability in a certain form. When working with probability, we often see activities such as rolling a die, tossing a coin, or selecting a ball from a bag. For example, we may be asked to find the probability of rolling a four on a die or the probability of selecting a blue ball from a bag.

Since these are practical activities and are referred to as experiments, we might think we have to carry them out in real life in order to calculate any probabilities. However, this is not necessarily the case. Theoretical probability describes the behavior we expect to happen in theory. To calculate the theoretical probability of an event, we need to be given a precise description of the system that gives rise to it. We then use logic to analyze the system and work out how it should behave in theory.

Before looking at some examples, we’ll first recall some basic terminology relating to experiments. The possible results of an experiment are known as outcomes. The sample space 𝑆 is the set of all possible outcomes. An event 𝐸 is a set of outcomes to which a probability is assigned. It is a subset of the sample space 𝑆 written as shown. And we use the term “favorable outcomes” to refer to the outcomes we want to test for in an experiment. For a given experiment, once we have identified the sample space and the event we want to test for, we can calculate the theoretical probability of that event using the following formula. The probability of an event is equal to the number of favorable outcomes divided by the total number of possible outcomes.

More formally, this can be written as follows. Let 𝑆 be the sample space; 𝐸, which is a subset of 𝑆, be the event; 𝑛 of 𝑆 be the number of elements in 𝑆; and 𝑛 of 𝐸 be the number of elements in 𝐸. Then, the probability of the event 𝑃 of 𝐸 is given by the formula 𝑃 of 𝐸 is equal to 𝑛 of 𝐸 over 𝑛 of 𝑆.

We will now consider some examples where we need to calculate the theoretical probability of an event.

A box contains five red marbles, eight green marbles, and four yellow marbles. If a marble is picked from the box at random, what is the probability that the marble is red?

We begin by considering the information given in the question. We have a box containing red, green, and yellow marbles. There are five red marbles, eight green marbles, and four yellow marbles. And we are asked to find the probability that a randomly selected marble is red. This is an example of theoretical probability. And we recall that the probability of an event can be written as a fraction where the numerator is the number of favorable outcomes and the denominator is the total number of possible outcomes. In this question, there are five red marbles. Since five plus eight plus four is equal to 17, there are 17 marbles in total. And we can therefore conclude that the probability of selecting a red marble is five out of 17 or five seventeenths.

In our next example, we need to find the probability of an event in a dice experiment.

What is the probability of rolling a number greater than or equal to four on a fair die?

We begin by recalling that a regular fair die has six faces numbered from one to six. This means that when we roll the die, there are six possible outcomes. In this question, we are interested in the event of rolling a number greater than or equal to four. There are three ways of satisfying this, rolling a four, rolling a five, or rolling a six. We recall that we can write the probability of an event as a fraction, where the numerator is the number of favorable outcomes and the denominator the total number of possible outcomes.

In this question, the probability of rolling a number greater than or equal to four is three out of six or three-sixths. Since the numerator and denominator are divisible by three, we can simplify this fraction. And we can therefore conclude that the probability of rolling a number greater than or equal to four on a fair die is one-half. It is worth noting that we could also write our answer as a decimal or percentage. One-half is equal to 0.5 and 50 percent.

In our next example, we’re asked to calculate the probability of an event without knowing the size of the sample space.

A bag contains an unknown number of balls. Given that one-sixth of the balls are white, one-fifth of them are green, and the rest are blue, what is the probability that a ball drawn at random from the bag is blue?

In this question, we are told that a bag contains three different colored balls. They are white, green, and blue. We are not told the number of balls that are each color or the total number of balls in the bag. We are told, however, that one-sixth of the balls are white. And this means that the probability of selecting a white ball at random is one-sixth. Likewise, we are told that one-fifth of the balls are green, so the probability of selecting a green ball is one-fifth. It is the probability of selecting a blue ball, that we will call 𝑥, that we are trying to calculate.

We recall that the sum of the probabilities of all outcomes in a sample space must equal one. This means that in this question the probability of selecting a white ball plus the probability of selecting a green ball plus the probability of selecting a blue ball must equal one. Substituting in the values we know, we have one-sixth plus one-fifth plus 𝑥 is equal to one. On the left-hand side of our equation, we can begin by adding one-sixth and one-fifth by finding the lowest common denominator, which is equal to 30. One-sixth is equivalent to five thirtieths and one-fifth is equivalent to six thirtieths.

Adding the numerators, we see that one-sixth plus one-fifth is equal to eleven thirtieths. Our equation simplifies to eleven thirtieths plus 𝑥 is equal to one. Noting that one whole one is equal to thirty thertieths. We can subtract eleven thirtieths from both sides. This gives us 𝑥 is equal to nineteen thirtieths. And we can therefore conclude that the probability of selecting a blue ball from the bag is nineteen thirtieths.

In some questions, rather than calculating probabilities of events, we are given the probability and have to work backwards using the formula to find the size of the set of outcomes corresponding to a particular event. This is true of our next example.

A bag contains 30 colored marbles. The probability of choosing a white marble at random is two-fifths. How many white marbles are in the bag?

We begin by recalling that the probability of an event is equal to the number of favorable outcomes divided by the total number of possible outcomes. This can be written more formally as shown. 𝑃 of 𝐸 is equal to 𝑛 of 𝐸 over 𝑛 of 𝑆, where 𝑆 is a sample space and 𝑛 of 𝑆 is its size, 𝐸 is the event we’re interested in, and 𝑛 of 𝐸 is its size and 𝑃 of 𝐸 is its theoretical probability.

In this question, the probability of selecting a white marble is equal to the number of white marbles divided by the total number of marbles. We are told in the question that there are 30 marbles in total, and two-fifths of these are white. Letting the number of white marbles be 𝑥, we have two-fifths is equal to 𝑥 over 30. We can multiply both sides of this equation by 30 such that 𝑥 is equal to two-fifths multiplied by 30 or two-fifths of 30. As one-fifth of 30 is equal to six, two-fifths of 30 is 12. We can therefore conclude that there are 12 white marbles in the bag.

An alternative method would be to notice that the fractions two-fifths and 𝑥 over 30 must be equivalent. And since five multiplied by six is equal to 30 and two multiplied by six is equal to 12, the number of white marbles must be equal to 12.

We will now consider one final example involving theoretical probability.

In a class of 50 students, 33 passed the mathematics test and 31 passed the language test. What is the probability that a randomly selected student failed in the language test?

In questions of this type, it is important that we’re able to select the relevant information. We are only interested in what happened in the language test. This means that the fact we are told that 33 students passed the mathematics test is irrelevant. We are told that there are 50 students in total. Of these, 31 passed the language test. Since 50 minus 31 is equal to 19, we know that 19 students failed the language test. And it is the probability of this event that we are trying to calculate.

We know that the probability of an event can be written as a fraction. It is the number of favorable outcomes divided by the total number of possible outcomes. The probability that a randomly selected student failed in the language test is therefore equal to 19 over 50, which could also be written as the decimal 0.38 or 38 percent.

We will now finish this video by summarizing the key points.

When given a description of an experiment, we need to identify its sample space — that is, the set of all possible outcomes — and the event we want to measure, the set of favorable outcomes. The theoretical probability of such an event is given by the following formula. The probability of an event is equal to the number of favorable outcomes divided by the total number of possible outcomes. More formally, this can be written as shown, where 𝑆 is the sample space; 𝐸, which is a subset of 𝑆, is the event; 𝑛 of 𝑆 is the number of elements in 𝑆; 𝑛 of 𝐸 the number of elements in 𝐸; and 𝑃 of 𝐸 is the probability of the event 𝐸.

We saw in this video that we can work backward from a given probability using the formula to find the size of the set of outcomes corresponding to a particular event. Finally, we saw that we can sometimes calculate a missing theoretical probability even if we do not know the size of the sample space. This is because the theoretical probability of an event is also the proportion of the total number of outcomes that are favorable.