Lesson Explainer: Theoretical Probability | Nagwa Lesson Explainer: Theoretical Probability | Nagwa

Lesson Explainer: Theoretical Probability Mathematics • 7th Grade

In this explainer, we will learn how to interpret a data set by finding and evaluating the theoretical probabilities.

In probability, when an event is the result of an experiment with a single stage, we say that this is a simple event. So, for example, tossing a coin once is a single stage experiment. The coin landing with the heads facing up is a simple event.

To calculate the probability of a simple event, we need to know the number of ways we can get a favorable outcome (i.e., the number of ways the thing we want to happen can happen) and the total number of all possible outcomes (i.e., the number of things that could possibly happen).

The probability of a simple event 𝐸 occurring is then 𝑃(𝐸)=.numberofwaysafavorableoutcomecanoccurtotalpossibleoutcomes

In our coin tossing example, if our simple event is “getting a heads” in a single throw, then the probability of getting a heads is 𝑃()=12=0.5heads.

There is one way to get a heads and only two possible outcomes altogether: heads or tails.

Let us look at some examples.

Example 1: Simple Probability: Random Selection of a Game

When Sarah gets home from school every day, she likes to play computer games for a while before starting her homework. She currently has 8 games to choose from and has no particular favorite.

If Sarah selects her game for today at random, find the probability that she plays Cool Cats.

Answer

Sarah has 8 games in total: Hamster Hike, Sid III, Foggy Farm, Cool Cats, Meer Mountain, Renaissance Star II, Strike Replay, and Quasar V, only one of which is Cool Cats.

The probability that Sarah selects Cool Cats is, therefore, 𝑃()==18=0.125.CoolCatsnumberofwaysofselectingCoolCatstotalnumberofgames

If Sarah chooses a game at random, then the probability that she selects Cool Cats to play today is therefore 0.125. If we convert this to a percentage (0.125×100=12.5%), we can say that Sarah has a 12.5% chance of selecting Cool Cats.

Example 2: Probability of Simple Events on a Construction Site

For safety reasons, color-coded helmets are worn by different types of workers on construction sites. On each site, a chart is marked up with the number of each type of worker present every working day. The table below shows the number of each type of worker present last Tuesday on the site for Garden Towers II. Find the probability that a worker chosen at random last Tuesday was a supervisor.

Answer

To find the probability that a worker chosen at random was a supervisor, we need first to find the total number of workers. We can then put the number of supervisors over this in a fraction for the required probability.

Let us mark the total on our table, together with the number of supervisors.

The total number of workers last Tuesday was 4+18+1+15=38supervisorssignallerssitemanagervisitors/other. Out of these, there were 4 supervisors. The probability that a worker chosen at random was a supervisor is, therefore, 𝑃()==438=438=0.1053.supervisornumberofsupervisorstotalnumberofworkerstod.p

Hence, the probability that a worker chosen at random was a supervisor is 0.105 to 3 decimal places. If we multiply this by 100 (0.105×100=10.5), we can say that 10.5% of the workforce on site last Tuesday were supervisors.

Example 3: Simple Events: Rolling a Die

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

Answer

To find the probability that we score an odd number with one throw of a die, we need to know how many odd numbers we could get out of the six faces.

There are three odd-numbered faces: 1, 3, and 5. Therefore, the probability that we score an odd number is 𝑃()==36=12=0.5.oddnumberofoddfacestotalnumberoffaces

Example 4: Simple Probability: Selecting Students at Random

A class contains 6 boys and 21 girls. If a pupil is selected at random, what is the probability they are a girl?

Answer

Since there are 21 girls and 6 boys in the class, there are 21+6=27 students in total. If a pupil is selected at random, then the probability that pupil is a girl is 𝑃()=2127=79=0.78girlto2d.p.

As a percentage, 0.78 is 0.78×100=78%. Hence, we can say that there is a 78% chance that a student selected at random is a girl.

In our final example, we will consider the difference between a simple and a compound event.

Example 5: Comparing Simple and Compound Events

If a regular six-sided die is rolled twice, what is the probability of getting a 3 on the first roll?

Answer

The probability of getting a 3 on the first roll of a die is the number of ways we can get a 3 with one roll of a die, divided by the number of possible outcomes. Since there is one way to get a 3 and there are 6 possible outcomes, the probability of getting a 3 is 16.

Note

The fact that the die is rolled a second time has no bearing at all on what happens in the first roll.

In fact, regardless of what happened in the first throw, as a separate, simple event, the probability of rolling a 3 on the second throw is also 16.

The situation would change, however, if we were to ask, for example, “what is the probability of rolling a 3 twice?” (i.e., rolling a 3 and then another 3). In this case, we are combining two experiments for a particular outcome (3 and 3 again) into a compound event and we are no longer talking about a simple event. In this case, we have to multiply our probabilities together: 𝑃(33)=𝑃(3)×𝑃(3)=16×16=136.

Finally, let us summarize the key points relating to simple events.

Key Points

  • A simple event is an event that results from a single-stage experiment. To calculate the probability of a simple event, we need the following:
    • The number of ways we can get a favorable outcome, that is, the number of ways the thing we want to happen can happen.
    • The total number of all possible outcomes, that is, the number of things that could possibly happen (which we call the sample space).
  • The probability of a simple event 𝐸 occurring is 𝑃(𝐸)=𝐸.numberofwaysfavorableoutcome()canoccurtotalpossibleoutcomes

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