14.03 Theoretical probability
We can make predictions for trials by first creating the sample space and then determining the theoretical probability of each outcome.
Exploration
If you roll two six-sided dice and add the numbers together, what is the probability of getting a sum of $6$6? What about a sum of $10$10 or greater? $8$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:
| Second die | |||||||
| $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 | ||
| First die | $1$1 | $1,1$1,1 | $1,2$1,2 | $1,3$1,3 | $1,4$1,4 | $1,5$1,5 | $1,6$1,6 |
| $2$2 | $2,1$2,1 | $2,2$2,2 | $2,3$2,3 | $2,4$2,4 | $2,5$2,5 | $2,6$2,6 | |
| $3$3 | $3,1$3,1 | $3,2$3,2 | $3,3$3,3 | $3,4$3,4 | $3,5$3,5 | $3,6$3,6 | |
| $4$4 | $4,1$4,1 | $4,2$4,2 | $4,3$4,3 | $4,4$4,4 | $4,5$4,5 | $4,6$4,6 | |
| $5$5 | $5,1$5,1 | $5,2$5,2 | $5,3$5,3 | $5,4$5,4 | $5,5$5,5 | $5,6$5,6 | |
| $6$6 | $6,1$6,1 | $6,2$6,2 | $6,3$6,3 | $6,4$6,4 | $6,5$6,5 | $6,6$6,6 | |
We can now tell that there are $36$36 possible outcomes. Depending on the trial we can highlight the favourable outcomes corresponding to the event, and the probability of any particular event is given by the formula
$\text{Probability}=\frac{\text{Number of favourable outcomes}}{36}$Probability=Number of favourable outcomes36
Explore this applet to find the various probabilities:
Once we have a sample space with every outcome being equally likely, we can express the probability as a fraction, decimal, or percentage.
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.
Worked example
A letter is chosen at random from the word "MATHEMATICS" two hundred times. and the results written down in a list.
How many times can we expect that a "T" will be chosen? Round your answer to the nearest whole number.
Think: There are $11$11 letters in the word "MATHEMATICS", and each one is equally likely to be chosen. We can think of the sample space like this spinner:
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Once we know the probability of "T" in one trial, we can multiply it by $200$200 to find the expected number of times that "T" will appear in the list.
Do: Using the formula,
$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}=\frac{2}{11}$Probability=Number of favourable outcomesTotal number of outcomes=211
This means each time a letter is chosen, there is a $\frac{2}{11}$211 chance of choosing a "T".
Multiplying this probability by the number of trials ($200$200) tells us the expected number of times that "T" will be chosen:
$\text{Number of Ts}=200\times\frac{2}{11}=\frac{400}{11}$Number of Ts=200×211=40011
As a decimal this is $36.\overline{36}$36.36, which rounds to the nearest whole number $36$36. This is how many times we should expect "T" to be chosen in the list of $200$200 letters.
Unless the question tells you to round your answer, you should not round it.
The exact value of theoretical probability is often very important, so we won't approximate it with a percentage or decimal unless we are told to do so.
Practice questions
Question 1
A two-digit number is formed using the numbers $3$3 and $2$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?
What is the probability that the number formed is more than $30$30?
Question 2
An eight-sided die is rolled $25$25 times. How many times should we expect to roll a $7$7? Round your answer to the nearest whole number.
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| eight-sided die |
Question 3
A bag contains $28$28 red marbles, $27$27 blue marbles, and $26$26 black marbles.
What is the probability of drawing a blue marble?
A single trial is drawing a marble from the bag, writing down the colour, and putting it back. If this trial is repeated $400$400 times, how many blue marbles should you expect?
Round your answer to the nearest whole number.