P - Probability Lesson

Probability

Probability measures the likelihood of an event occurring. This measure is based on the total number of outcomes possible and can be expressed as a fraction or a decimal from 0-1, or a percent from 0-100%.

As you can see by the number line below, if the probability is close to 0, the event is unlikely. If the probability is close to 1, the event is very likely. When the probability is near equation image indicator or 0.5, then that event is as likely as not. Probability of an event is written: P (event).

probability number line: impossible, unlikely, as likely as not, likely, certain

Example

Chelle is going to select a random card from 12 cards. They are numbered 1 to 12. What is the probability that she will choose a card with a number less than 4? Determine if this event is likely or unlikely.

Count the number of favorable outcomes.

Numbers less than 4: 1, 2, 3

Count the number of total outcomes.

Find the probability P(number less than 4)

3/12 or 1/4

Determine if this event is likely or unlikely.

The probability of choosing a card with a number less than 4 is unlikely. (less than $LaTeX: \frac{1}{2}$ $\frac{1}{2}$ )

Example

A bag contains 7 red marbles, 3 purple marbles, and 2 blue marbles. What is the probability of picking a blue marble?

Count the number of favorable outcomes

2 blue marbles

Count the total number of outcomes

12 marbles

Find the probability P(blue marble)

2/12 or 1/6

The probability of P(blue marble) is 1/6 or unlikely.

Let's try a few more:

image of dice Determine whether each of the following events is impossible, unlikely, as likely as not, likely, or certain.

- Roll an odd number on a fair number cube (1-6).

Solution: Half of the numbers are odd and half are even; P(odd) is as likely as not

- Roll a number less than 3 on a fair number cube (1-6)

Solution: There are only 2 numbers less than 3: 1 and 2. $LaTeX: \frac{2}{6}=\frac{1}{3}$ $\frac{2}{6}=\frac{1}{3}$ ; P(<3) is unlikely

- Roll a number equal to or greater than 3 on a fair number cube (1-6)

Solution: There are 4 numbers in this scenario: 3, 4, 5, 6. $LaTeX: \frac{4}{6}=\frac{2}{3}$ $\frac{4}{6}=\frac{2}{3}$ ; P≥ is likely.

- Roll a number less than 7 on a fair number cube (1-6)

Solution: All numbers are less than 7. $LaTeX: \frac{6}{6}=1$ $\frac{6}{6}=1$ ; P(<7) is certain.

- Roll a number greater than 6 on a fair number cube (1-6)

Solution: None of the numbers are greater than 6. 0. P(>6) is impossible.

Complementary Events

Now, what about those outcomes that are not the expected or favorable event. These events are called complementary events. In other words, the sum of the event and its complement will always equal 1.

P(event) + P(complement) = 1

Let's add the complement to our practice from above

1. Roll an odd number on a fair number cube (1-6).

Solution: P(odd)= $LaTeX: \frac{1}{2}$ $\frac{1}{2}$
Complement is P(no odd)
$LaTeX: \frac{1}{2}$ $\frac{1}{2}$ ; $LaTeX: \frac{1}{2}+\frac{1}{2}=1$ $\frac{1}{2}+\frac{1}{2}=1$

2. Roll a number less than 3 on a fair number cube (1-6).

Solution: P(less than 3)= $LaTeX: \frac{1}{3}$ $\frac{1}{3}$
Complement is P(3 or greater)= $LaTeX: \frac{2}{3}$ $\frac{2}{3}$
$LaTeX: \frac{1}{3}+\frac{2}{3}=1$ $\frac{1}{3}+\frac{2}{3}=1$

3. Roll a number equal to or greater than 3 on a fair number cube (1-6).

Solution: $LaTeX: P\left(\ge3\right)=\frac{2}{3}$ $P\left(\ge3\right)=\frac{2}{3}$
Complement is $LaTeX: P\left(<3\right)=\frac{1}{3}$ $P\left(<3\right)=\frac{1}{3}$
$LaTeX: \frac{2}{3}+\frac{1}{3}=1$ $\frac{2}{3}+\frac{1}{3}=1$

4. Roll a number less than 7 on a fair number cube (1-6).

Solution: $LaTeX: P\left(<7\right)=1$ $P\left(<7\right)=1$
Complement is $LaTeX: P\left(>7\right)=0$ $P\left(>7\right)=0$
$1+0=1$

5. Roll a number greater than 6 on a fair number cube (1-6).

Solution: $LaTeX: P\left(>6\right)=0$ $P\left(>6\right)=0$
Complement is $LaTeX: P\left(\le6\right)=1$ $P\left(\le6\right)=1$
$0+1=1$

Probability Word Problems

1. Bailey's math teacher almost always introduces a new topic by showing a short video clip. Bailey's class just completed a topic on Friday. Should she expect the teacher to show a video clip next week?

Solution: Since they finished studying a topic, they will be learning about a new topic. Since the teacher almost always starts off with a video clip, it is likely that the teacher will show a video clip next week.

2. Mike's music club meets on Wednesday afternoons. How likely is it that Mike is at the movies on Wednesday afternoon?

Solution: Since the music club meets on Wednesdays, it is unlikely that Mike is at the movies on Wednesday afternoon.

3.Teri rides the bus to school if she gets up by 7:30 A.M. Teri gets up at 7:30 A.M. about half of the time so can you estimate the probability that Teri will ride the bus to school?

Solution: The probability that she will ride the bus is ½, so it is just as likely as not that she will ride the bus to school.

Watch the following video to see more examples of probability.

Probability Practice

Probability Homework

Now that you have spent some time learning how to describe the likelihood of an event, and write it as a fraction, decimal, or percent, you are ready to complete your Probability: Probability Homework. Download your homework by CLICKING HERE. Links to an external site.

Once you have completed your homework, AND MAKE SURE YOU ATTEMPTED AND WORKED THE PROBLEMS OUT ON YOUR OWN, click here to download your homework key. Links to an external site.

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