If you roll two six-sided dice and add the numbers together, what is the probability of getting a sum of 6? What about a sum of 10 or greater? 8 or less?

Before we answer these questions we need to determine the sample space. The possible outcomes for two dice can be drawn in a grid:

	1	2	3	4	5	6
1	1, 1	1, 2	1, 3	1, 4	1, 5	1, 6
2	2, 1	2, 2	2, 3	2, 4	2, 5	2, 6
3	3, 1	3, 2	3, 3	3, 4	3, 5	3, 6
4	4, 1	4, 2	4, 3	4, 4	4, 5	4, 6
5	5, 1	5, 2	5, 3	5, 4	5, 5	5, 6
6	6, 1	6, 2	6, 3	6, 4	6, 5	6, 6

We can now tell that there are 36 possible outcomes. Depending on the trial we can highlight the favorable outcomes corresponding to the event, and the probability of any particular event is given by the formula:\text{Probability} = \dfrac{\text{Number of favorable outcomes}}{36}

Exploration

Explore this applet to find the various probabilities:

Loading interactive...

Once we have a sample space with every outcome being equally likely, we can express the probability as a fraction, decimal, or percentage.

Examples

Example 1

A two-digit number is formed using the numbers 3 and 2. It can be two of the same or one of each number in any order.

What is the probability that the number formed is odd?

Worked Solution

Create a strategy

Count the numbers that are odd and use the probability formula.

Apply the idea

Four different numbers can be formed: 22,\,23,\,32 and 33. A two-digit number is odd if its units digit is odd. We have 2 numbers with an odd unit digits out of 4.

\displaystyle \text{Probability}	\displaystyle =	\displaystyle \frac{2}{4}	Substitute the values
	\displaystyle =	\displaystyle \frac{1}{2}	Simplify

What is the probability that the number formed is more than 30?

Worked Solution

Create a strategy

Count the number that has 3\,as a Tens digit and use the probability formula.

Apply the idea

Four different numbers can be formed: 22,\,23,\,32 and 33. We have 2 numbers with a 3 as the tens digit out of 4.

\displaystyle \text{Probability}	\displaystyle =	\displaystyle \frac{2}{4}	Substitute the values
	\displaystyle =	\displaystyle \frac{1}{2}	Simplify

Idea summary

To find the theoretical probability of an event use the formula: \text{Theoretical probability}=\dfrac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}

Repeating trials

Once we know the probability of an event, we can predict how many times this event will occur if a trial is repeated several times.

We multiply the probability of the event by the number of trials, rounding to the nearest whole number.

Examples

Example 2

An eight-sided die is rolled 25 times. How many times should we expect to roll a 7? Round your answer to the nearest whole number.

An 8-sided die. Ask your teacher for more information.

Worked Solution

Create a strategy

Find the probability then multiply it by the number of trials.

Apply the idea

The probability of rolling a 7 is \dfrac {1}{8}.

\displaystyle \text{Number of times}	\displaystyle =	\displaystyle \frac {1}{8} \times 25	Multiply by the number of trials
	\displaystyle =	\displaystyle \frac{25}{8}	Evaluate
	\displaystyle =	\displaystyle 3	Round to a whole number

So, we can expect to roll a 7, \, \, 3 times.

Example 3

A bag contains 28 red marbles, 27 blue marbles, and 26 black marbles.

A bag contains a 28 red marbles, 27 blue marbles, and 26 black marbles.

What is the probability of drawing a blue marble?

Worked Solution

Create a strategy

We should divide the number of blue marbles by the total number of marbles to find the probability.

Apply the idea

There are 27 blue marbles in a total of 28+27+26= 81 marbles.

\displaystyle \text{Probability}	\displaystyle =	\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}	Use the formula
	\displaystyle =	\displaystyle \frac{27}{81}	Substitute the values

A single trial is drawing a marble from the bag, writing down the colour, and putting it back. If this trial is repeated 400 times, how many blue marbles should you expect? Round your answer to the nearest whole number.

Worked Solution

Create a strategy

Multiply the probability of drawing a blue marble in a single trial by the number of trials.

Apply the idea

\displaystyle \text{Number of blue marble}	\displaystyle =	\displaystyle \frac{27}{81} \times 400	Multiply the probability by the number of trials
	\displaystyle =	\displaystyle \frac{400}{3}	Evaluate into one fraction
	\displaystyle =	\displaystyle 133	Round to the nearest whole number

So, we should expect 133 blue marbles in a 400 trials.

Idea summary

To predict how many times an event will occur if a trial is repeated several times, we multiply the probability of the event by the number of trials.

12.03 Theoretical probability

Ideas

Theoretical probability

Exploration

Examples

Example 1

Repeating trials

Examples

Example 2

Example 3

Outcomes

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