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Definition of Probability

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Have you ever been in a talent show? Take a look at this dilemma.

Since the opening of J.S. Middle School, a tradition has been the end of the year talent show. The school opened ten years ago, and within that time there have been 8 talent shows. There were two years when the school was not able to host one because there was flooding or repairs were being done in auditorium.

“I wonder if we are going to have the talent show this year,” Carmen asked at lunch one day.

“I am sure that we are,” Tyler said biting into his ham sandwich. “After all, there were only two years that the talent show did not happen and that was because of the circumstances.”

“Well, are there any circumstances this year?”

“I don’t think so. The probability is high that it is going to happen.”

“What is the probability of the talent show happening?” Carmen asked taking a sip of milk.

Before Tyler answers, let’s think about this. What is the probability of the event happening?

To answer this question, you need to know about probability. This Concept will teach you how to figure out probability by thinking about favorable outcomes and total outcomes. By the end of the Concept, you will know how to answer this question.

Guidance

Probability is something that you hear about all the time. Anytime you talk about the chances that something will or won’t happen, you are talking about probability. The trick about probability is that it isn’t just about talking. It is also about math. There are mathematical ways of figuring out the likelihood that an event is going to or not going to occur. But let’s start with an understanding of probability.

What is probability?

Probability is the likelihood that an event will occur. It is a mathematical way of calculating how likely an event is likely to occur. An event is a result of an experiment or activity that might include such things as:

  • flipping a coin
  • spinning a spinner
  • rolling a number cube
  • choosing an item from a jar or bag

How do we calculate probability?

We calculate probability by looking at the ratio of favorable outcomes to total outcomes in a given situation. In ratio form, the probability of an event is:

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

Write this ratio down in your notebook.

An outcome is a possible result of some event occurring. For example, when you flip a coin, “heads” is one outcome; tails is a second outcome. Total outcomes are computed simply by counting all possible outcomes.

Keep in mind as you go through this Concept that all outcomes used are presumed to be “fair.” That is – when you flip a coin, the outcomes of heads or tails are equally likely. When you spin a spinner, sections are all of equal size and equally likely to be landed on. When you toss a number cube, faces of the cube are the same size and again equally likely to be landed on. And so on.

For flipping a coin:

\text{total outcomes} & = \text{heads, tails}\\ & = 2 \ \text{total outcomes}

For tossing a number cube:

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

For selecting a day of the week:

\text{total outcomes} & = \text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\\& = 7 \ \text{total outcomes}

Favorable outcomes are the specific outcomes you are looking for.

For flipping a coin and having it come up heads:

\text{favorable outcomes} & = \text{heads, tails}\\& = 1 \ \text{favorable outcome}

For tossing a number cube and having it come with up an even number:

\text{favorable outcomes} & = \cdot \ 1 \ \cdot \cdot \ 2 \ \cdot \cdot \cdot \ 3 \ \cdot \cdot \cdot \cdot \ 4 \ \cdot \cdot \cdot \cdot \cdot \ 5 \ \cdot \cdot \cdot \cdot \cdot \cdot \ 6\\& = 3 \ \text{favorable outcomes}

For randomly choosing a date and have it land on a weekday:

\text{favorable outcomes} & = \text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\\& = 5 \ \text{favorable outcomes}

To write a ratio, we compare the favorable outcome to the total outcomes. Comparing favorable outcomes to possible total outcomes is what we call theoretical probability.

Now that you know how to think about theoretical probability, we can look at calculating some simple probabilities. Remember, that we are going to be writing ratios that compare favorable outcomes with total outcomes.

The probability of any event is written as P (\text{event}) . So:

P(A) is the probability that event A will occur.

P (\text{heads}) is the probability that heads will turn up on a flipped coin.

P (5) is the probability that a number cube will turn up as 5

Here is another one.

What is the probability of flipping heads on a coin?

To work through this probability, we are going to be writing a ratio that compares the number of favorable outcomes with the total number of outcomes.

Favorable outcome = 1 since there is one heads on a coin

Total outcomes = 2 since there is the possibility of heads or tails

The answer is 1:2.

For tossing a number cube and having it land an even number:

Our final answer is 1:2.

To find any probability, follow the steps below.

Problem : What is the probability of the arrow landing on a yellow section?

Step 1 : Count the number of favorable outcomes.

There are 2 yellow spaces, so

favorable outcomes = 2

Step 2 : Count the number of total outcomes.

There are 5 spaces in all, so

total outcomes = 5

Step 3 : Write the ratio of favorable outcomes to total outcomes

P (\text{yellow}) &= \text{favorable outcomes} : \text{total outcomes}\\&= 2:5

Take a few minutes to write these steps in your notebook.

Write each theoretical probability. Be sure to simplify the ratio when necessary.

Example A

What is the probability of rolling a 1 or a 3 on a number cube?

Solution: 2:6 or 1:3

Example B

If there are four blue marbles and one red marble in a bag, what is the probability of pulling out a red one?

Solution: 1:5

Example C

What is the probability of pulling out a blue one?

Solution: 4:5

Here is the original problem once again. Reread it and then answer the following questions about probability.

Since the opening of J.S. Middle School, a tradition has been the end of the year talent show. The school opened ten years ago, and within that time there have been 8 talent shows. There were two years when the school was not able to host one because there was flooding or repairs were being done in auditorium.

“I wonder if we are going to have the talent show this year,” Carmen asked at lunch one day.

“I am sure that we are,” Tyler said biting into his ham sandwich. “After all, there were only two years that the talent show did not happen and that was because of the circumstances.”

“Well, are there any circumstances this year?”

“I don’t think so. The probability is high that is going to happen.”

“What is the probability of the talent show happening?” Carmen asked taking a sip of milk.

To think about the probability of the talent show happening, we can take the data from the past ten years and create a ratio.

\text{Probability} = \frac{favorable \ outcomes}{total \ outcomes}

In the past ten years, there have been 8 talent shows. There have been ten possible years to calculate with. These are the total outcomes. Here is our ratio.

8:10

Vocabulary

Probability
the likelihood that an event will happen.
Event
result of an experiment or an activity.
Favorable Outcome
the outcome that you are looking for.
Total Outcome
the total number of possible outcomes.
Ratio
a comparison of two quantities.
Theoretical Probability
the ratio that compares the number of favorable outcomes to the number of total outcomes.

Guided Practice

Here is one for you to try on your own.

What is the probability of the arrow landing on a silver or pink section?

Answer

Step 1 : Count the number of favorable outcomes.

There are 2 silver spaces and 1 pink space, so

favorable outcomes = 3

Step 2 : Count the number of total outcomes.

There are 5 spaces in all, so

total outcomes = 5

Step 3 : Write the ratio of favorable outcomes to total outcomes

P (\text{silver or pink}) = \text{favorable outcomes} : \text{total outcomes} = 3:5

This is our answer.

Video Review

This is a Khan Academy video on probability.

Practice

Directions : Answer each question or solve each problem as it connects to probability.

For rolling a 4 on the number cube.

1. List each favorable outcome.

2. Count the number of favorable outcomes.

3. Write the total number of outcomes.

For rolling a number greater than 2 on the number cube:

4. List each favorable outcome.

5. Count the number of favorable outcomes.

6. Write the total number of outcomes.

For rolling a 5 or 6 on a number cube:

7. List each favorable outcome.

8. Count the number of favorable outcomes.

9. Write the total number of outcomes.

A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with an even number on it:

10. List each favorable outcome.

11. Count the number of favorable outcomes.

12. Write the total number of outcomes.

A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with a number greater than 3:

13. List each favorable outcome.

14. Count the number of favorable outcomes.

15. Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be orange.

16. Count the number of favorable outcomes.

17. Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be large:

18. Count the number of favorable outcomes.

19. Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be blue:

20. Count the number of favorable outcomes.

21. Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be small:

22. Count the number of favorable outcomes.

23. Write the total number of outcomes.

For randomly choosing a marble and having it turn out to be orange and large:

24. Count the number of favorable outcomes.

25 Write the total number of outcomes.

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