Lesson Explainer: Probability of Simple Events Mathematics

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

The probability of an event is the likelihood of it occurring.

When we discuss the likelihood of an event happening in everyday life, we may use some common words to describe this likelihood, for example, “certain”, “likely”, “very unlikely”, or “impossible”. In mathematics, we can assign a numerical value to a probability. Impossible events have a probability of 0, and events that are certain to happen have a probability of 1. Events that are equally likely can be written with a probability of 0.5, or 12.

The sum of the probabilities of all possible outcomes must equal 1. For example, when flipping a coin, the probability of getting “heads” plus the probability of getting “tails” is 1. This is because the probability of getting either one of heads or tails is certain, that is, a probability of 1.

In probability terms, a simple event refers to an event with a single outcome, for example, getting “heads” with a single toss of a coin, or rolling a 4 on a die.

We also need to consider “fairness” when discussing probability.

Definition: Fair Experiments

A probability experiment is considered as fair if all outcomes are equally likely.

An experiment where the outcomes are not equally likely is unfair or biased.

Consider the situation of flipping a fair coin. It can be described as fair as the outcomes are equally likely. If the coin has sides “heads” and “tails”, then the outcome of getting “tails” would be 1 outcome out of a possible 2 outcomes. We could write this as a fraction, 12.

Definition: Probability of a Simple Event

The probability of a simple event is probabilityofaneventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes=.

Commonly in probability, we may use the notation 𝑃()event to represent the probability of an event occurring. For example, when selecting a green or blue ball from a bag, 𝑃()green can be used to represent the probability of selecting a green ball.

We can now see how this information can be applied in a number of different examples.

Example 1: Determining the Theoretical Probability of an Event

A class has 18 boys and 9 girls. What is the probability that a randomly selected student is a girl?

Answer

We can recall that the probability of a simple event can be written as probabilityofaneventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes=. In this case, we need to calculate the probability of selecting a girl, which we can write as 𝑃()girl.

We can write the statement 𝑃()=.girlnumberofgirlstotalnumberofstudents

As there are 18 boys and 9 girls in the class, then the total number of students must be 18+9=27. Substituting the information that the number of girls = 9 and the total number of students = 27 gives us 𝑃()=927.girl

Simplifying this fraction, we have 𝑃()=13.girl

Thus, the probability that a randomly selected student is a girl is 13.

In the following example, we will see how we may often need to use physical information about an object to obtain the likelihood of an event happening, for example, by examining the sections of a spinner.

Example 2: Determining the Probability of an Event Involving a Spinner

What is the probability of the pointer landing on an even number when the given spinner is spun?

Answer

We consider that in this spinner, as the sections are of equal size, then there is an equal probability of the spinner landing in each section, assuming that the spinner is fair.

We recall that the probability of a simple event is given by probabilityofaneventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes=.

To find the probability of landing on an even number, 𝑃()even, we can write 𝑃()=.evennumberofevenvaluesonthespinnertotalnumberofsectionsonthespinner

We consider the even and odd values on the spinner. Even numbers are integers that are divisible by 2. Odd numbers are integers that are not divisible by 2. As 12 and 14 are the only even numbers on the spinner, then the number of possible outcomes that are even is 2. The total number of outcomes is the total number of sections on the spinner, 8. We can substitute these values into our equation, giving 𝑃()=28.even

Simplifying the fraction, we have 𝑃()=14.even

Therefore, the probability of the pointer landing on an even number when the spinner is spun is 14.

We consider another example.

Example 3: Determining the Probability of an Event

A deck of cards contains cards numbered from 1 to 81. If a card is picked at random, what is the probability of picking a card with a number that is divisible by 5?

Answer

We can consider the deck of cards as follows.

In order to find the probability of picking a particular card or type of card, we recall that the probability of a simple event can be given as probabilityofaneventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes=.

For the event of picking a card that is divisible by 5, 𝑃(5)divisibleby, we could write the equation that 𝑃(5)=5.divisiblebynumberofcardvaluesdivisiblebytotalnumberofcards

We recall that divisible means to be able to divide by a number and get an integer answer. Numbers that are divisible by 5 are also multiples of 5. We can list the numbers that are divisible by 5, between 1 and 81, as 5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80.and As the highest card value is 81, then there are no higher possible values. Counting these values, we see that there are 16.

Next, as there are 81 cards, then the total number of cards is 81.

Filling these values into the equation above gives 𝑃(5)=1681.divisibleby

We cannot simplify this fraction any further. Therefore, the probability of of picking a card with a number divisible by 5 from this deck of cards is 1681.

We will now look at an example where we are given information about the total number of outcomes and the probability of an event to work out the number of outcomes of a specific event.

Example 4: Using Theoretical Probability to Solve a Problem

There are 28 people in a meeting. The probability that a person chosen at random is a man is 12. Calculate the number of women in the meeting.

Answer

The question gives us the value that the probability of choosing a man from the total number of people in the room is 12. We can use this information, along with the information about the total number of people, to find the number of men in the room.

Recall that the probability of a simple event can be given as probabilityofaneventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes=.

For this scenario, we can write the probability of picking a man, 𝑃()man, as 𝑃()=.mannumberofmeninthemeetingtotalnumberofpeopleinthemeeting

Given that 𝑃()=12man and the totalnumberofpeopleinthemeeting=28, we can substitute these values into the equation above, giving 12=28.numberofmen

Multiplying both sides of this equation by 28 and simplifying gives us 28×12=14=.numberofmennumberofmen

Since the number of men in the meeting is 14, then we can calculate the number of women by subtracting the number of men, 14, from the total number of people, 28. As 2814=14,

the number of women in the meeting is 14.

We will now look at an example where we find the probability of picking an even digit from a given number.

Example 5: Determining the Probability of an Event

If a single digit is selected at random from the number 224, 839, 287, what is the probability of the digit being even?

Answer

Recall that the probability of a simple event is given by probabilityofaneventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes=.

In this question, the probability of picking an even digit, 𝑃()evendigit, can be found by 𝑃()=.evendigitnumberofevendigitstotalnumberofdigits

Considering each digit separately and determining its parity, we have

Counting the number of even digits above, there are 6 even values. The total number of digits is 9. Substituting these values into the probability equation above gives us 𝑃()=69=23.evendigit

Therefore, the probability of selecting an even digit from the number 224, 839, 287 is 23.

In the final example, we will use a given probability and information about the specific number of an outcome, to work out the total number of outcomes.

Example 6: Using Theoretical Probability to Solve a Problem

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

We are given the number of white balls in the bag, and we are given the probability of picking one of the unknown number of red balls as 731. To find the total number of balls, we can apply the probability equation for finding the probability of a simple event: probabilityofaneventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes=.

To find the number of white balls, we can recognize that since there are only red or white balls in the bag and the total of all of the probabilities is 1, then the probability of getting white, 𝑃()white, can be found by 𝑃()=1𝑃()=1731=2431.whitered

The probability of picking a white ball can be found by 𝑃()=.whitenumberofwhiteballstotalnumberofballs

Substituting in the values 𝑃()=2431white and the number of white balls = 24 gives us 2431=24.totalnumberofballs

Rearranging the equation by cross multiplying the fractions, we have 24×=31×24.totalnumberofballs

Dividing both sides of the equation by 24 gives totalnumberofballs=31.

We can check our answer by calculating that the number of red balls must be 3124=7. Thus, the probability of picking a red ball would be the given value of 731, since 𝑃()==731.rednumberofredballstotalnumberofballs

This confirms our answer; the total number of balls in the bag is 31.

We now summarize the key points.

Key Points

  • The probability of an event is the likelihood of it occurring.
  • The sum of the probabilities of all possible outcomes must equal 1.
  • A probability experiment is considered as fair if all outcomes are equally likely. Experiments that are unfair are often referred to as biased.
  • A simple event is an event with a single outcome.
  • The probability of a simple event, 𝑃()event, is 𝑃()=.eventnumberofoutcomeswheretheeventoccurstotalnumberofpossibleoutcomes

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