23.04 Theoretical probability
Probability with numbers
Now that we know how to describe events with language, we will now investigate using numbers to calculate probabilities. If we can split up the sample space into equally likely outcomes and can identify the favourable outcomes making up an event, we can use the formula:
$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Probability=Number of favourable outcomesTotal number of outcomes
Worked example
Example 1
Hunter has $11$11 black shirts, $9$9 white shirts, $4$4 yellow shirts and $3$3 red shirts. If he selects a shirt from his closet at random, what is the probability he selects either a yellow shirt or a red shirt?
Think: Each individual shirt is an outcome. They are all equally likely to be picked, so we can use the equation:
$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Probability=Number of favourable outcomesTotal number of outcomes
The event we are thinking about is "he selects either a yellow shirt or a red shirt", so the number of favourable outcomes is the number of red shirts plus the number of yellow shirts. The total number of outcomes is the total number of shirts.
Do: The number of favourable outcomes is $4+3=7$4+3=7.
The total number of outcomes is $11+9+4+3=27$11+9+4+3=27.
The probability will be $\frac{7}{27}$727.
We can also use a useful fact about complementary events - since exactly one of them must happen, their probabilities always add to $1$1. This means if we know the probability of an event, the probability of the complementary event will be one minus the probability of the original:
$\text{Probability of complementary event}=1-\text{Probability of event}$Probability of complementary event=1−Probability of event
Worked example
Example 2
Ursula selects a card at random from a standard deck. What is the probability she will select a card that isn't a king?
Think: There are $13$13 different card values in a standard deck of cards. The event "selecting a card that isn't a king" is the complementary event to "selecting a card that is a king". We can calculate the probability of the complementary event and subtract it from $1$1 to find our probability.
Do: The probability of selecting a king is $\frac{1}{13}$113, so the probability of selecting a card that isn't a king will be $1-\frac{1}{13}=\frac{12}{13}$1−113=1213.
Reflect: We could calculate this probability another way, by thinking about the $12$12 card values other than the king as "favourable outcomes", and dividing by the total number of card values ($13$13).
If a sample space can be split up into equally likely outcomes, then we can use the following formula:
$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Probability=Number of favourable outcomesTotal number of outcomes
If two events are complementary then their probabilities will add to $1$1. This means:
$\text{Probability of complementary event}=1-\text{Probability of event}$Probability of complementary event=1−Probability of event
Practice questions
Question 1
Consider this list of numbers:
$2,2,2,3,3,3,4,4,5,5,5,7,7,7,7,9,9$2,2,2,3,3,3,4,4,5,5,5,7,7,7,7,9,9
How many numbers are in the list?
A number is chosen from the list at random. What is the probability it is an odd number?
What number has the same probability of being picked as $4$4?
A number is chosen from the list at random. What number has the highest probability of being chosen?
Question 2
The probability of the local football team winning their grand final is $0.36$0.36.
What is the probability that they won't win the grand final?
Question 3
A standard deck of $52$52 cards is shown below.
If a card is selected at random, what is the probability that it is:
a red card?
a card between $5$5 and $9$9 inclusive?
a card that is red and has a number between $5$5 and $9$9 inclusive?
a card that is red or a king?