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Lesson Explainer: Probability of Simple Events Mathematics

In this explainer, we will learn how to find the probability of a simple event and write it as a fraction.

Probability in math is concerned with measuring the chance or likelihood of something happening. In day-to-day life, we often encounter situations where we use the language of probability. For example, if you are planning a day trip to the beach tomorrow, you might want to know what the likelihood of rain is tomorrow. If you buy a lottery ticket, you might not want to know how tiny the likelihood of winning is! If you toss a fair coin, you know that you have a 50–50 chance of landing tails.

Probabilities can be written as fractions, decimals, or ratios. In this explainer, we will look at probabilities as fractions.

Let us look at an example.

Example 1: Probability as a Fraction

What is the probability of rolling a number greater than 5 on a fair die?

Answer

There are 6 faces on a fair die.

There is only one face with more than 5 dots on it. That is the face with 6 dots.

So, we have a 1 in 6 chance of rolling a number greater than 5 on a fair die. As a fraction, we write this as follows: theprobabilityofrollinganumbergreaterthan5onafairdie=16.

Before we try some further examples, let us look at this idea more closely and remind ourselves of some of the key ideas associated with probability.

Key Ideas: Probabilities as Fractions

  • To write the probability of a particular outcome as a fraction, we need to know two quantities:
    1. the total number of possible outcomes,
    2. the number of “favorable” outcomes.
    A “favorable” outcome is one that fits our requirements, that is, the particular outcome we are looking for.
    The probability of a particular outcome (or set of outcomes) as a fraction is then probabilitynumberoffavorableoutcomesnumberofallpossibleoutcomes=. We can restate this in slightly more mathematical language: for event 𝐴, the probability that 𝐴 occurs is 𝑃(𝐴)=𝑛(𝐴)𝑛(𝑆), where 𝑛(𝐴) is the number of elements in the sample space that correspond to event 𝐴 occurring and 𝑛(𝑆) is the total number of possible outcomes in the sample space.
    Recall that a sample space of an experiment or trial is the set of all possible outcomes for that experiment. An experiment in probability is any procedure that has a specific set of outcomes and that can be repeated.
  • Since all probabilities are less than or equal to 1, the denominator of the fraction must be larger than (or equal to) the numerator.
  • The sum of the probabilities of all possible outcomes in an experiment is equal to 1. So, the sum of the probabilities of all distinct events in the sample space is equal to 1.

Example 2: Probabilities as Fractions

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

Answer

To find the probability that the score on a roll with a regular six-sided die is an even number, we need to know how many possible outcomes from the roll could land an even number.

There are 3 faces of a regular die with an even number of dots (2, 4, and 6), and we know that the total number of faces (i.e., the total possible outcomes on a roll of the die) is 6. The probability of rolling an even number is, therefore, probabilityofanevennumbernumberoffaceswithanevennumbertotalnumberoffaces==36=12.

Let us look at another example, this time using numbered cards.

Example 3: Probabilities as Fractions

A card is drawn from a deck of cards numbered from 1 to 40. What is the probability that the number on the card is odd?

Answer

To find the probability that the number on a card drawn from a pack of 40 cards is odd, we need to first find how many odd-numbered cards there are in the pack. If we strike out all of the even-numbered cards, we will be left with only odd, so we can then count these up.

There are 20 odd-numbered cards (highlighted in yellow) and 40 cards altogether, so the probability of choosing an odd-numbered card is probabilityofchoosinganoddcardnumberofoddcardstotalnumberofcards==2040=12.

Now let us look at an example with more mathematical elements.

Example 4: Probability of 𝑥 in Fractions

Denote by 𝑥 the number rolled on a fair die. What is the probability, on a single roll, that 3<𝑥<6?

Answer

If 𝑥 is the number rolled, to find the probability that 3<𝑥<6, we must determine how many of the possible outcomes are greater than 3 but less than 6.

We can see from the diagram above that there are two possibilities, which have been circled in red: the numbers 4 and 5. The probability required is then 𝑃(3<𝑥<6)=3<𝑥<6=26=13.numberoffaceswithdotstotalnumberoffaces

In our next example, we will use probability as a fraction to gain information from the results of a survey.

Example 5: Gaining Information Using Probabilities as Fractions

The table shows the results of a survey to find out how many students use various methods of transportation to travel to school. If a student is selected at random, what is the probability that they walk to school?

Answer

To find the probability that a student selected at random walks to school, we need to know first how many students walk to school and then how many students were surveyed in total.

From the table, we can see that there were 100 students in total, of which 52 go to school on foot. So if a student was selected at random, the probability that they walk to school is 𝑃()==52100.studentswhowalknumberofstudentswhowalktotalnumberofstudentssurveyed

In fact, in this case, it might be helpful to convert our fraction into a decimal or a percentage: 52100=0.52 which, multiplied by 100 for the percentage, is 52%. There is, therefore, a 52% chance that a student chosen at random walks to school.

Let us look at one more example of how probabilities as fractions can help us gain information from a table of data.

Example 6: Using Probabilities as Fractions to Gain Information from Data

The table shows the number of sixth, seventh, and eighth graders in a school. If one of these students is randomly selected, find 𝑃 (sixth grader or seventh grader). Give your answer as a fraction.

Sixth GradersSeventh GradersEighth Graders
282141

Answer

If a student is chosen at random, to find 𝑃 (sixth grader or seventh grader), we must find how many sixth and seventh graders there are and divide this by the total number of students.

The number of sixth and seventh graders is 28+21=49, and the total number of students is 28+21+41=90. Hence, the probability that a student chosen at random is either a sixth grader or a seventh grader is 𝑃()==4990.sixthgraderorseventhgradernumberofsixthandseventhgraderstotalnumberofstudents

Finally , let us remind ourselves of the key ideas associated with probabilities as fractions.

Key Points

  • To write the probability of a particular outcome as a fraction, we need to know
    • the total number of possible outcomes,
    • the number of “favorable” outcomes.
    A “favorable” outcome is one that fits our requirements, that is, the particular outcome we are looking for.
    The probability of a particular outcome (or set of outcomes) as a fraction is then probabilitynumberoffavorableoutcomesnumberofallpossibleoutcomes=. We can restate this in slightly more mathematical language: for event 𝐴, the probability that 𝐴 occurs is 𝑃(𝐴)=𝑛(𝐴)𝑛(𝑆), where 𝑛(𝐴) is the number of elements in the sample space that correspond to event 𝐴 occurring and 𝑛(𝑆) is the total number of possible outcomes in the sample space.
    Recall that a sample space of an experiment or trial is the set of all possible outcomes for that experiment. An experiment in probability is any procedure that has a specific set of outcomes and that can be repeated.
  • Remember that all probabilities are less than or equal to 1, so the denominator of the fraction must be larger than (or equal to) the numerator.
  • Recall also that the sum of the probabilities of all possible outcomes in an experiment is equal to 1. So, the sum of the probabilities of all distinct events in the sample space is equal to 1.

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