Range of Probabilities and Complementary Events Revision Notes for HSC SSCE Mathematics Standard

Range of Probabilities and Complementary Events

Understanding the range of probabilities

Probability is a numerical measure of how likely an event is to occur. This value is always constrained to a specific range. When an event cannot possibly occur, we assign it a probability of 0. When an event is guaranteed to happen, we assign it a probability of 1.

All probability values must fall within this range, meaning they can be anywhere from $0$ to $1$ , including these endpoints. We express this mathematically as:

$0 \leq P(E) \leq 1$

This can also be written as two separate conditions: $P(E) \geq 0$ and $P(E) \leq 1$ .

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The probability range is inclusive of both endpoints, meaning events can have probabilities of exactly 0 or exactly 1. This is indicated by the "less than or equal to" symbols ( $\leq$ ) rather than strict inequality symbols.

The probability scale

Think of probability as a scale with key positions:

$0$ : Impossible – the event cannot happen
$0.25$ : Unlikely but possible
$0.5$ : Even chance – equally likely to happen or not happen
$0.75$ : Likely to happen
$1$ : Certain – the event will definitely happen

chatImportant

It is not possible to have a probability value of $2$ or any number greater than $1$ , just as you cannot have a negative probability. Any probability calculation that gives a result outside the range $0$ to $1$ indicates an error in your working.

The sum of probabilities rule

When you consider all possible events or outcomes in an experiment, their probabilities must add up to exactly $1$ . This is because one of these outcomes must occur – they cover all possibilities.

We write this as:

$P(A) + P(B) + P(C) + ... = 1$

where $A$ , $B$ , $C$ , ... represent all the possible outcomes or events in the experiment.

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This is sometimes called the "completeness" rule – the probabilities must be "complete" and sum to $1$ because we've accounted for every possible outcome. Nothing is missing, and nothing is counted twice.

Calculating probabilities within the range

When some probabilities are known, we can use the sum rule to find unknown probabilities. This is particularly useful when dealing with multiple outcomes.

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Worked Example: Using the Range of Probability

Question: A box contains red, yellow and blue cards. The probability of selecting a red card is $\frac{3}{5}$ and the probability of selecting a yellow card is $\frac{1}{10}$ . What is the probability of selecting a blue card?

Solution:

Step 1: Write the formula for the range of probability.

$P(R) + P(Y) + P(B) = 1$

Step 2: Substitute the known probabilities into the formula.

We know that $P(R) = \frac{3}{5}$ and $P(Y) = \frac{1}{10}$

$\frac{3}{5} + \frac{1}{10} + P(B) = 1$

Step 3: Solve the equation by making $P(B)$ the subject.

$P(B) = 1 - \frac{3}{5} - \frac{1}{10}$

Step 4: Simplify the fraction.

To subtract these fractions, we need a common denominator of $10$ :

$P(B) = 1 - \frac{6}{10} - \frac{1}{10}$

$P(B) = \frac{10}{10} - \frac{6}{10} - \frac{1}{10}$

$P(B) = \frac{3}{10}$

Step 5: Write the answer in words.

The probability of selecting a blue card is $\frac{3}{10}$ .

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Checking Your Answer

Always verify that your answer makes sense. The probability should be between $0$ and $1$ , and when you add all three probabilities together, they should equal $1$ :

$\frac{3}{5} + \frac{1}{10} + \frac{3}{10} = \frac{6}{10} + \frac{1}{10} + \frac{3}{10} = \frac{10}{10} = 1$ ✓

Understanding complementary events

A complementary event includes all the possible outcomes that are NOT part of the original event. The complement gives us a useful alternative way to calculate probabilities.

What is a complement?

For any event $E$ , its complement (written as $\overline{E}$ and pronounced "E bar") consists of all outcomes in the sample space that are not in $E$ .

Example: When throwing a die, if event $E$ is "rolling a 2", then the complement $\overline{E}$ is "rolling a 1, 3, 4, 5, or 6".

The complementary probability rule

When you combine an event with its complement, you've covered every possible outcome. Since one of these must occur, they are certain to happen together. This means their probabilities sum to $1$ :

$P(E) + P(\overline{E}) = 1$

We can rearrange this formula to find the probability of a complement:

$P(\overline{E}) = 1 - P(E)$

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Key Notation

$E$ represents the event or outcome we're interested in
$\overline{E}$ represents the complement of event E, or all outcomes not including event $E$

When to use complements

Using the complement formula is particularly helpful when:

It's easier to calculate the probability of something NOT happening
You're asked to find "not" probabilities
The event you want has many outcomes, but its complement has few

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Strategy for Using Complements

If you see the word "not" in a probability question, this is a strong signal to use the complement formula. It's almost always faster to calculate $1 - P(E)$ than to count all the outcomes that satisfy "not $E$ ".

Calculating complementary probabilities

The complement formula provides an efficient method for solving certain probability problems. Rather than counting all the favourable outcomes, sometimes it's quicker to count the unfavourable ones and subtract from $1$ .

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Worked Example: Using the Complementary Event

Question: Lisa selects a card at random from a normal pack. Find the probability of obtaining the following outcomes:

a Not a 10

b Not a black jack (i.e. not a jack of clubs or spades)

c Not a picture card

Solution:

Part a: Not a 10

Step 1: Write the formula for the complement.

$P(\overline{10}) = 1 - P(10)$

Step 2: Substitute the probability for a 10.

In a standard deck of $52$ cards, there are four 10s (one in each suit), so $P(10) = \frac{4}{52}$

$P(\overline{10}) = 1 - \frac{4}{52}$

Step 3: Evaluate.

$P(\overline{10}) = \frac{52}{52} - \frac{4}{52}$

$P(\overline{10}) = \frac{48}{52}$

Step 4: Simplify the fraction.

$P(\overline{10}) = \frac{12}{13}$

Answer: The probability of not selecting a 10 is $\frac{12}{13}$ .

Part b: Not a black jack

Step 5: Write the formula for the complement.

$P(\overline{\text{black jack}}) = 1 - P(\text{black jack})$

Step 6: Substitute the probability for a black jack.

There are two black jacks in a deck (jack of clubs and jack of spades), so $P(\text{black jack}) = \frac{2}{52}$

$P(\overline{\text{black jack}}) = 1 - \frac{2}{52}$

Step 7: Evaluate.

$P(\overline{\text{black jack}}) = \frac{52}{52} - \frac{2}{52}$

$P(\overline{\text{black jack}}) = \frac{50}{52}$

Step 8: Simplify the fraction.

$P(\overline{\text{black jack}}) = \frac{25}{26}$

Answer: The probability of not selecting a black jack is $\frac{25}{26}$ .

Part c: Not a picture card

Step 9: Write the formula for the complement.

$P(\overline{\text{picture}}) = 1 - P(\text{picture})$

Step 10: Substitute the probability for a picture card.

Picture cards are jacks, queens, and kings. There are $3$ picture cards in each of $4$ suits, giving 12 picture cards total. So $P(\text{picture}) = \frac{12}{52}$

$P(\overline{\text{picture}}) = 1 - \frac{12}{52}$

Step 11: Evaluate.

$P(\overline{\text{picture}}) = \frac{52}{52} - \frac{12}{52}$

$P(\overline{\text{picture}}) = \frac{40}{52}$

Step 12: Simplify the fraction.

$P(\overline{\text{picture}}) = \frac{10}{13}$

Answer: The probability of not selecting a picture card is $\frac{10}{13}$ .

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Card Deck Reference

In card problems, remember that a standard deck has 52 cards with 4 suits. Picture cards are jacks, queens, and kings (not aces). Always simplify your final answer to its lowest terms.

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Key Points to Remember:

Probability range: All probabilities must fall between $0$ and $1$ inclusive, expressed as $0 \leq P(E) \leq 1$
Sum to one: The probabilities of all possible outcomes in an experiment must add up to exactly $1$
Complementary events: The complement $\overline{E}$ includes all outcomes NOT in event E, and together they cover all possibilities
Complement formula: Use $P(\overline{E}) = 1 - P(E)$ to find the probability of an event's complement
Strategy tip: When calculating "not" probabilities, the complement formula is usually faster than counting all the favourable outcomes directly

Range of Probabilities and Complementary Events (HSC SSCE Mathematics Standard): Revision Notes