Lesson Explainer: Probability and Simple Events | Nagwa Lesson Explainer: Probability and Simple Events | Nagwa

Lesson Explainer: Probability and Simple Events Mathematics • Second Year of Preparatory School

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

We begin by recapping some of the key terminology associated with probability.

Key Terms: Probability

Experiment: An experiment is an activity with an identifiable result.

Outcome: An outcome is a specific result of an experiment.

Sample space: A sample space is the set of all possible outcomes from a random experiment.

Event: An event is a subset of the sample space. An event may consist of a single outcome or may be composed of multiple outcomes.

Let us consider the experiment of rolling a six-sided die and recording the number. Here, a specific outcome would be rolling a particular number, such as 4. The sample space would be the set of all outcomes, in this case {1,2,3,4,5,6}, and an event would be a subset of the sample space (which could be single or multiple outcomes), for example, “rolling a 1” or “rolling a number greater than 3.”

Informally, we may think of an event as “something that happens.” The probability of an event is how likely it is to happen. We may discuss probabilities in our everyday lives using words like certain, impossible, unlikely, or very likely. For example, we may say, “it is likely that it will rain tomorrow.” Here, the event is it raining tomorrow, and the probability we are ascribing to it is “likely.”

In mathematics, we want to extend this idea and start describing probabilities using numbers. We define impossible events as having a probability of 0, and events that are certain to happen as having a probability of 1. This leads to the probability scale, on which the probabilities of all events are placed between these values.

We describe events that have a probability of 0.5 as having an “even chance” of occurring. They are just as likely to happen as they are not to happen. Events with a probability between 0 and 0.5 are less likely to occur, and events with a probability between 0.5 and 1 are more likely to occur. We might also use terms like highly unlikely or highly likely to describe events for which the probabilities are close to the extremes of 0 and 1. This is illustrated on the probability scale below.

Probabilities can be expressed either as decimals, fractions, or percentages. We can also express probabilities as ratios, but this is less common.

When all the outcomes of an experiment are equally likely, the probability of an event can be calculated by dividing the number of outcomes in that event, which we term the number of “successful” outcomes, by the total number of outcomes in the sample space: probabilityofaneventnumberofsuccessfuloutcomestotalnumberofoutcomes=.

We can formalize this in the definition below.

Definition: The Probability of an Event

If 𝐴 is an event in a sample space 𝑆, where each outcome is equally likely, then the probability of event 𝐴 occurring is 𝑃(𝐴)=𝑛(𝐴)𝑛(𝑆), where 𝑃(𝐴) represents the probability of event 𝐴, 𝑛(𝐴) represents the number of elements in event 𝐴, and 𝑛(𝑆) represents the number of elements in the sample space 𝑆.

We will now consider a series of examples. In each example, the key will be to determine the total number of outcomes (or elements) in the sample space of the experiment and the number of outcomes that are considered to be “successful.”

Example 1: Calculating the Probability of Rolling a Number on a Fair Six-Sided Die

If I roll a regular six-sided die, what is the probability that the score is 3?

Answer

As the die is regular, this means that it is unbiased, and so every outcome is equally likely. To calculate the required probability, we can use the formula probabilityofaneventnumberofsuccessfuloutcomestotalnumberofoutcomes=.

Here, the event we are interested in is getting a 3 when a regular six-sided die is rolled. The possible outcomes from rolling such a die are {1,2,3,4,5,6}, and so the total number of outcomes is 6.

The only successful outcome is getting the number 3, so there is one successful outcome. Hence, 𝑃(3)=16.

In our next example, we will calculate the probability of a simple event that consists of more than one outcome.

Example 2: Calculating the Probability of a Simple Event

If I roll a regular six-sided die, what is the probability that the score is divisible by 3?

Answer

As the die is regular, this means that it is unbiased, and so every outcome is equally likely. To calculate the required probability, we can use the formula probabilityofaneventnumberofsuccessfuloutcomestotalnumberofoutcomes=.

Here, the event we are interested in is getting a score that is divisible by 3 when a regular six-sided die is rolled. Therefore, we can express this probability as 𝑃(3)=3.divisiblebynumberofoutcomesthataredivisiblebytotalnumberofoutcomes

The possible outcomes from rolling such a die are {1,2,3,4,5,6}, and so the total number of outcomes is 6.

The possible outcomes that are divisible by 3 are {3,6}, and hence the number of successful outcomes is 2. Therefore, 𝑃(3)=26=13.scoreisdivisibleby

As we saw in the previous example, if we are giving probabilities as fractions, it is generally a good practice to give them in their simplest forms, canceled down as far as possible.

We summarize the steps involved in calculating the probability of a simple event below.

How To: Calculating the Probability of a Simple Event

  1. Determine the total number of equally likely outcomes. This is the same as determining the number of elements in the sample space.
  2. Determine how many of these are considered “successful” outcomes.
  3. Divide the number of successful outcomes by the total number of outcomes.

Example 3: Calculating the Probability of Selecting a Prime Number from a Deck of Numbered Cards

A card is drawn at random from a deck of cards numbered 1 to 52. What is the probability that the card drawn is a prime number?

Answer

We are asked to determine the probability that a card drawn at random from a deck of cards is a prime number. This is not a usual deck of playing cards, but rather a set of 52 numbered cards, each bearing one of the integers from 1 to 52 inclusive. The card is to be drawn at random, and so every card has an equal chance of being chosen. To calculate the required probability, we can use the formula 𝑃(𝐴)=𝑛(𝐴)𝑛(𝑆), where 𝑃(𝐴) represents the probability of event 𝐴, 𝑛(𝐴) represents the number of outcomes in event 𝐴, and 𝑛(𝑆) represents the number of elements in the sample space 𝑆.

So, 𝑛(𝑆)=52, as there are 52 cards in the deck.

𝑛(𝐴), in this case, is the number of prime numbers between 1 and 52. We recall that a prime number is a number that has exactly two distinct factors: 1 and itself. The number 1 is not a prime number as it has only one distinct factor (1). To determine if a number is prime, we can consider whether it is divisible by every integer less than or equal to its square root. For some divisors, such as 2 and 5, this is easily done, but to check whether other numbers are factors, we may need to perform a short division or consider our multiplication tables.

The prime numbers between 1 and 52 are

  • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.

Hence, 𝑛(𝐴)=15. Using the formula above, we find that 𝑃()=1552.prime

In our next example, we will calculate an experimental probability from survey data that has been presented in a frequency table.

Example 4: Calculating a Probability given a Frequency Table

The table shows the results of a survey that asked 100 people to vote for their favorite type of TV program.

DramaDocumentaryComedyNewsSport
14 19 14 16 37

What is the probability that a randomly selected person prefers drama?

Answer

The data have been presented in the form of a frequency table. The top row of the table lists the different types of TV programs, and the second row gives the frequency for each type. As the person is to be selected at random, each person has an equal chance of being chosen, and so we can calculate the required probability using the formula probabilityofaneventnumberofsuccessfuloutcomestotalnumberofoutcomes=.

In this context, the number of successful outcomes is the number of people who preferred drama, and the total number of outcomes is the total number of people surveyed: 𝑃()=.prefersdramanumberofpeoplewhopreferdramatotalnumberofpeople

We are told in the question that 100 people were surveyed. We can confirm this by summing the frequencies in the second row of the table: 14+19+14+16+37=100.

From the first column of the table, we identify that the number of people who said they preferred drama is 14. Hence, 𝑃()=14100=750.prefersdrama

We can use probabilities to make inferences, which are conclusions based on statistical data. For example, in the previous problem, we found that the probability that a randomly selected person prefers drama was 14100 or 14%. In contrast, the probability that a randomly selected person prefers sport is 37100 or 37%. Based on this information, the person responsible for the programming schedule may choose to show a greater proportion of sports programs than drama programs in order to appeal to a larger number of people. It is therefore important that whenever samples are taken, they are unbiased and representative of the population being sampled, so that the decisions made as a result are based on reliable data.

It is also possible to work backward from knowing a probability to determine an unknown, such as the total number of outcomes or the number of successful outcomes. To do this, we will usually be required to form and solve an algebraic equation. We will now consider one final example in which we are given the probability of an event (choosing a red ball from a bag) and the number of balls of another color in the bag, and then we are asked to determine the total number of balls in the bag.

Example 5: Calculating an Unknown given the Probability of an Event

A bag contains 24 white balls and an unknown number of red balls. The probability of choosing a red ball at random is 731. How many balls are there in the bag?

Answer

There are a number of possible approaches to this question. We will demonstrate an algebraic method in which we form and solve an equation.

In order to determine the total number of balls in the bag, we will first determine the number of red balls. As the number of red balls is unknown, we can represent it using the letter 𝑟. The bag contains only red and white balls, and as there are 24 white balls in the bag, an expression for the total number of balls in the bag is 𝑟+24.

We are told that the probability of choosing a red ball at random is 731. We recall that probabilities of simple events can be calculated using the formula probabilityofaneventnumberofsuccessfuloutcomestotalnumberofoutcomes=.

In this context, we can rewrite this as 𝑃()=.rednumberofredballstotalnumberofballs

We can form an equation by substituting 731 for the probability of choosing a red ball, 𝑟 for the number of red balls, and (𝑟+24) for the total number of balls. Thus, 731=𝑟𝑟+24.

To solve this equation for 𝑟, we begin by cross-multiplying: 7𝑟+168=31𝑟.

To group like terms on the same side of the equation, we subtract 7𝑟 from each side giving 168=24𝑟.

Finally, we divide both sides of the equation by 24 to give 7=𝑟.

Hence, there are 7 red balls in the bag. As there are also 24 white balls in the bag, the total number of balls in the bag is 7+24=31.

Let us finish by recapping some key points.

Key Points

  • When all the outcomes of an experiment are equally likely, the probability of an event can be calculated using the following formula: probabilityofaneventnumberofsuccessfuloutcomestotalnumberofoutcomes=.
  • More formally, if 𝐴 is an event in a sample space 𝑆, where each outcome is equally likely, 𝑃(𝐴)=𝑛(𝐴)𝑛(𝑆), where 𝑃(𝐴) represents the probability of event 𝐴, 𝑛(𝐴) represents the number of outcomes in event 𝐴, and 𝑛(𝑆) represents the number of elements in the sample space 𝑆.

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