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Probability | Lesson

What is a probability?

A probability indicates the chance that an

will happen.

A probability can be any number from

0

1

A probability of $0$ ‍ or $0 %$ ‍ means that an event will never happen.
A probability of $\frac{1}{2}$ ‍ or $50 %$ ‍ means that an event is equally likely to happen or not happen.
A probability of $1$ ‍ or $100 %$ ‍ means that an event will certainly happen.

The probability that event

A

will happen is written as

P (A)

The probability that event

A

will not happen,

P (not A)

, is equal to

1 - P (A)

What skills are tested?

Calculating the probability that an event will happen (or will not happen) based on a description of the situation
Calculating the probability that an event will happen (or will not happen) based on information in a data display
Calculating a count or quantity from a probability
Comparing probabilities or drawing conclusions based on probabilities

How is the probability of an event calculated?

An event could be the outcome of any random process such as the toss of a fair coin, the roll of a fair number cube, or the random selection of an item from a group.

We use the notation

P (A)

to represent "the probability that event

A

will happen".

P (A) = \frac{ways A can happen}{possible outcomes}

Sammy is about to roll a fair, six-sided number cube numbered

1

through

6

. What is the probability that Sammy will roll a number that is a multiple of

3

P (multiple of 3) = \frac{multiples of 3}{possible outcomes}

$3$ ‍ and $6$ ‍ are the only multiples of $3$ ‍ on the number cube. The number of multiples of $3$ ‍ is $2$ ‍.
There are $6$ ‍ different numbers on the number cube. The number of possible outcomes is $6$ ‍.

P (multiple of 3) = \frac{2}{6} = \frac{1}{3}

The probability that Sammy will roll a number that is a multiple of

3

\frac{1}{3}

How is the probability of an event calculated using information in a data display?

When calculating the probability that a randomly selected item will have certain attribute(s), the information we need may be in a data display.

P (A) = \frac{ways A can happen}{possible outcomes}

In such cases,

The ways $A$ ‍ can happen is the number of items of type $A$ ‍.
The number of possible outcomes is the total number in the group.

Pairs of Jeans	Youth	Adults
None	$2$ ‍	$7$ ‍
$1$ ‍	$4$ ‍	$6$ ‍
$2$ ‍	$8$ ‍	$4$ ‍
$3$ ‍	$5$ ‍	$2$ ‍
More than $3$ ‍	$6$ ‍	$1$ ‍
Total	$25$ ‍	$20$ ‍

Rita asked shoppers at a local mall how many pairs of jeans they own and recorded the responses in the table above. If a shopper from this survey is randomly selected, what is the probability that the selected shopper will be someone who owns

3

pairs of jeans?

Here, the event is "selecting a shopper who owns

3

pairs of jeans".

There are

5 + 2 = 7

shoppers in the survey who own

3

pairs of jeans.

A total of

25 + 20 = 45

shoppers were surveyed.

\begin{aligned} P (owns 3 pairs) & = \frac{owns 3 pairs}{total surveyed} \\ = \frac{7}{45} \end{aligned}

If a shopper from this survey is randomly selected, the probability that the shopper selected will be someone who owns

3

pairs of jeans is

\frac{7}{45}

How can we calculate a count or quantity from a probability?

Sometimes we are given the probability and are asked to calculate either the number of possible outcomes or the number of ways

A

can happen

To solve these questions:

Substitute known values into the equation $P (A) = \frac{ways A can happen}{possible outcomes}$ ‍.
Solve for the unknown quantity.

A bag contains a number of solid-colored marbles, of which

3

are blue,

5

are yellow, and the rest are red. If a person were to draw a marble at random from the bag, the probability that the marble drawn would be blue is

15 %

. How many marbles are in the bag?

\begin{aligned} P (blue) & = \frac{blue marbles}{total marbles} \\ \frac{15}{100} & = \frac{3}{total marbles} \end{aligned}

Notice that the numerator

15

is divided by

5

to get the numerator of the equivalent fraction. To complete the equivalent fraction we need to divide the denominator

100

5

\frac{15 \div 5}{100 \div 5} = \frac{3}{20}

Therefore, the total number of marbles must equal

20

There are

20

marbles in the bag.

Your turn!

TRY: CALCULATING A PROBABILTY

A box contains

5

red,

10

green, and

20

orange candies. If one candy is selected randomly from the box, what is the probability that the selected candy will be red?

TRY: CALCULATING A QUANTITY FROM A PROBABILITY

A gumball machine contains red, blue, and green gumballs. There are

70

red gumballs and

50

blue gumballs. When the handle of the machine is turned, one gumball is randomly dispensed. If the probability of getting a red gumball is

\frac{1}{3}

, how many green gumballs are in the machine?

TRY: COMPARING PROBABILITIES

Doughnut Variety	Number of Doughnuts
Cream filled	$16$ ‍
Plain	$22$ ‍
Jelly filled	$12$ ‍

The table above shows the number of doughnuts in a bakery display case by variety. If a doughnut is chosen at random from the display case, which of the following statements is true?

TRY: CALCULATING A PROBABILITY USING A TABLE

	Man	Woman	Child	Total
First class	$175$ ‍	$144$ ‍	$6$ ‍	$325$ ‍
Second class	$168$ ‍	$93$ ‍	$24$ ‍	$285$ ‍
Third class	$462$ ‍	$165$ ‍	$79$ ‍	$706$ ‍
Total	$805$ ‍	$402$ ‍	$109$ ‍	$1,316$ ‍

The table above shows data about the passengers aboard the Titanic. If a passenger who traveled on the Titanic is selected at random, what is the probability that the selected passenger was someone who traveled in first class?

Round your answer the nearest integer.

%

Things to remember

A probability indicates the chance that an event will happen.

A probability can be any number between

0

and

1

The probability that event A will happen is:

P (A) = \frac{ways A can happen}{possible outcomes}

The probability that event A will not happen is:

P (not A) = 1 - P (A)

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Praxis Core Math

Course: Praxis Core Math > Unit 1