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Empirical Probability

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Theoretical Probability

Have you ever thought about spinning a spinner to calculate probability? Take a look at this dilemma.

Jessie has a spinner that is divided into four colors: blue, red, yellow and green. She knows that the spinner can be used to calculate theoretical probability, but she can't remember what theoretical probability is. She is wondering how it is different from experimental probability.

Do you know?

Pay attention and you will learn all about theoretical probability is this Concept.

Guidance

Probability is defined as a mathematical way of calculating how likely an event is to occur.

The probability of an event occurring is defined as the ratio of favorable outcomes to the number of possible equally likely total outcomes in a given situation.

In ratio form, the probability of an event is:

P \text{(event)}=\text{favorable outcomes} : \text{total outcomes}

Theoretical probability is probability that is based on an ideal situation.

For instance, since a flipped coin has two sides and each side is equally likely to land up, the theoretical probability of landing heads (or tails) is exactly 1 out of 2. Whether or not the coin actually lands on heads (or tails) 1 out of every 2 flips in the real world does not affect theoretical probability. The theoretical probability of an event remains the same no matter how events turn out in the real world.

Take a look at this situation.

Find the probability of tossing a number cube and having it come up “4”.

Step 1: Find the total number of outcomes

\text{Total outcomes} &= \bullet \ 1 \ \bullet \bullet \ 2 \ \bullet \bullet \bullet \ 3 \ \bullet \bullet \bullet \bullet \ 4 \ \bullet \bullet \bullet \bullet \bullet \ 5 \ \bullet \bullet \bullet \bullet \bullet \bullet \ 6 \\&= 6 \ \text{total outcomes}

Step 2: Find the number of favorable outcomes.

\text{favorable outcomes} &= \bullet \ 1 \ \bullet \bullet \ 2 \ \bullet \bullet \bullet \ 3 \ \ {\color{red}\bullet \bullet \bullet \bullet \ 4} \ \bullet \bullet \bullet \bullet \bullet \ 5 \ \bullet \bullet \bullet \bullet \bullet \bullet \ 6 \\&= 1 \ \text{favorable outcome}

Step 3: Find the ratio of favorable outcomes to total outcomes.

\text{Favorable}:\text{Total}= 1:6

A prediction is a reasonable guess about what will happen in the future.

Good predictions should be based on facts and probability.

Predictions based on theoretical probability: These are the most reliable types of predictions, based on physical relationships that are easy to see and measure and that do not change over time. They include such things as:

  • coin flips
  • spinners
  • number cubes

Calculate each example of theoretical probability.

Example A

What is the probability of tossing a number cube and having it come up a two or a three?

Solution: 2:6

Example B

What is the probability of tossing a number cube and having it come up even?

Solution: 3:6

Example C

What is the probability of tossing a number cube and having it come up less than 6?

Solution: 5:6

Now let's go back to the dilemma from the beginning of the Concept.

The spinner can be used to calculate theoretical probability when it is considered without an experiment. Looking at the spinner itself, we can calculate the chances of spinning one or more colors. We haven't done an experiment with the spinner yet, we simply are calculating the probability by looking at the spinner. Experimental probability can be calculated once the spinner has been spun once or a number of times.

Vocabulary

Probability
a mathematical way of calculating how likely an event is to occur.
Favorable Outcome
the outcome that you are looking for
Total Outcomes
all of the outcomes both favorable and unfavorable.
Theoretical Probability
probability based on an ideal situation relating favorable to total outcomes

Guided Practice

Here is one for you to try on your own.

Two number cubes are thrown. What is the theoretical probability of rolling a number greater than 8?

Solution

First, let's think about the range of number combinations that we can possibly roll on the number cubes.

We can roll numbers from 2 to 12. There are 11 possible outcomes.

How many are greater than 8?

The values greater than 8 are 9, 10, 11 and 12.

There are four values greater than 8.

The theoretical probability of rolling a number greater than 8 is 4:11.

Video Review

Probability Part 2

Practice

Directions: Solve each problem.

A spinner has five sections: purple, yellow, green, blue and red.

  1. Find the probability for the arrow landing on blue on the spinner:
    1. List each favorable outcome.
    2. Count the number of favorable outcomes.
    3. Write the total number of outcomes.
    4. Write the probability.
  2. Find the probability for the arrow landing on red or green on the spinner:
    1. List each favorable outcome.
    2. Count the number of favorable outcomes.
    3. Write the total number of outcomes.
    4. Write the probability.
  3. Find the probability for the arrow NOT landing on yellow on the spinner:
    1. List each favorable outcome.
    2. Count the number of favorable outcomes.
    3. Write the total number of outcomes.
    4. Write the probability.
  4. Find the probability for rolling a 3 or 4 on the number cube:
    1. List each favorable outcome.
    2. Count the number of favorable outcomes.
    3. Write the total number of outcomes.
    4. Write the probability.
  5. Find the probability for rolling greater than 2 on the number cube:
    1. List each favorable outcome.
    2. Count the number of favorable outcomes.
    3. Write the total number of outcomes.
    4. Write the probability.

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