Identities Revision Notes for Grade 12 NSC Matric Mathematics

Identities

Understanding probability identities is essential for solving complex probability problems. These mathematical rules help us calculate the likelihood of different combinations of events occurring. Let's explore the key identities that form the foundation of probability theory.

Understanding different types of events

Before diving into the identities, it's crucial to understand how events can relate to each other. Events can be independent, dependent, or mutually exclusive, and each type requires different approaches when calculating probabilities.

Independent events occur when the outcome of one event does not affect the outcome of another event. For example, when you flip a coin twice, the result of the first flip doesn't influence the second flip. Each flip has the same probability regardless of previous results.

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Independence is a key concept in probability. When events are independent, knowing the outcome of one event tells us nothing about the likelihood of another event occurring.

Dependent events occur when the outcome of one event influences the probability of another event. Consider choosing items from a lunchbox containing sandwiches and apples - after eating one item, fewer choices remain, making subsequent selections dependent on previous choices.

Mutually exclusive events are events that cannot happen at the same time. For instance, when rolling a single die, you cannot get both an even number and an odd number on the same roll.

infoNote

Think of mutually exclusive events as "either this or that, but not both." This distinction is crucial for applying the correct probability formulas.

The addition rule

The addition rule (also known as the sum rule) helps us find the probability that either event A or event B (or both) will occur. This fundamental identity connects the probabilities of individual events with their union and intersection.

The formula is: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

This rule accounts for the overlap between events. We subtract $P(A \text{ and } B)$ because when we add $P(A)$ and $P(B)$ , we count the intersection twice. The subtraction corrects this double-counting.

The addition rule for mutually exclusive events

When dealing with mutually exclusive events, the addition rule simplifies significantly. Since these events cannot occur simultaneously, their intersection is empty, meaning $P(A \text{ and } B) = 0$ .

The formula becomes: $P(A \text{ or } B) = P(A) + P(B)$

This simplified version is much easier to use when you've confirmed that events cannot happen together. Always verify that events are truly mutually exclusive before applying this rule.

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Critical Check: Before using the simplified addition rule, always verify that events are truly mutually exclusive. A common mistake is assuming events are mutually exclusive when they're not!

The complementary rule

The complementary rule is one of the most useful probability identities. It states that the probability of an event not occurring equals one minus the probability of the event occurring.

The formula is: $P(\text{not } A) = 1 - P(A)$

This rule works because event A and its complement (not A) are mutually exclusive and exhaustive - exactly one of them must occur. Since the total probability of all possible outcomes equals 1, we can easily find the complement's probability.

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The complementary rule is particularly useful when it's easier to calculate the probability of something not happening rather than calculating it directly.

The product rule for independent events

When events are independent, we use the product rule to find the probability that both events occur. Since independence means one event doesn't affect the other, we simply multiply their individual probabilities.

The formula for independent events is: $P(A \text{ and } B) = P(A) \times P(B)$

However, when events are dependent, this relationship doesn't hold: $P(A \text{ and } B) \neq P(A) \times P(B)$

Worked examples

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Worked Example 1: Testing Independence with Replacement

Consider drawing beads from a bag containing 3 yellow and 4 black beads. We draw a bead, record its colour, replace it, then draw again.

Step 1: Calculate individual probabilities

$P(\text{first bead yellow}) = \frac{3}{7}$
$P(\text{second bead black}) = \frac{4}{7}$

Step 2: Calculate joint probability

Since we replace the first bead, the composition remains unchanged for the second draw.

$P(\text{yellow first and black second}) = \frac{3}{7} \times \frac{4}{7} = \frac{12}{49}$

Step 3: Test for independence

Since $\frac{12}{49} = \frac{3}{7} \times \frac{4}{7}$ , these events are independent.

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Worked Example 2: Testing Dependence without Replacement

Now consider the same bag, but without replacement. We draw two beads consecutively.

Step 1: Calculate individual probabilities

$P(\text{first bead red}) = \frac{3}{8}$ (assuming 3 red, 5 green beads)
$P(\text{second bead green}) = \frac{5}{8}$

Step 2: Calculate joint probability

After removing one bead, only 7 remain. If the first was red, 2 red and 5 green remain.

$P(\text{red first and green second}) = \frac{3}{8} \times \frac{5}{7} = \frac{15}{56}$

Step 3: Test for independence

Since $\frac{15}{56} \neq \frac{3}{8} \times \frac{5}{8} = \frac{15}{64}$ , these events are dependent.

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Worked Example 3: Using the Addition Rule

A survey reveals: $P(\text{drinks tea}) = 0.5$ , $P(\text{drinks coffee}) = 0.4$ , $P(\text{drinks both}) = 0.1$ .

Find $P(\text{drinks tea or coffee})$ :

$P(\text{tea or coffee}) = P(\text{tea}) + P(\text{coffee}) - P(\text{tea and coffee})$

$P(\text{tea or coffee}) = 0.5 + 0.4 - 0.1 = 0.8$

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Worked Example 4: Applying the Complementary Rule

Given $P(\text{learner drinks cola}) = 0.3$ , $P(\text{learner drinks lemonade}) = 0.6$ , and $P(\text{drinks neither}) = 0.2$ .

Using the complementary rule:

$P(\text{drinks cola or lemonade}) = 1 - P(\text{drinks neither}) = 1 - 0.2 = 0.8$

Then using the addition rule:

$P(\text{drinks both}) = P(\text{cola}) + P(\text{lemonade}) - P(\text{cola or lemonade})$

$P(\text{drinks both}) = 0.3 + 0.6 - 0.8 = 0.1$

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Worked Example 5: Mutually Exclusive Events

Consider natural numbers less than 16. Let A = {even numbers} and B = {prime numbers}.

Step 1: List the events

A = {2, 4, 6, 8, 10, 12, 14}
B = {2, 3, 5, 7, 11, 13}

Step 2: Check for mutual exclusivity

Since both sets contain 2, they're not mutually exclusive.

Step 3: Apply the general addition rule

$P(A) = \frac{7}{15}$
$P(B) = \frac{6}{15}$
$P(A \text{ and } B) = \frac{1}{15}$ (only the number 2)
$P(A \text{ or } B) = \frac{7}{15} + \frac{6}{15} - \frac{1}{15} = \frac{12}{15}$

chatImportant

Common Exam Mistakes to Avoid

Just because events are mutually exclusive doesn't mean they're independent. These are completely different concepts:

Test mutual exclusivity by checking if $P(A \text{ and } B) = 0$
Test independence by checking if $P(A \text{ and } B) = P(A) \times P(B)$

Always identify the type of sampling (with or without replacement) as this determines whether events are independent or dependent.

Key formulas summary

Addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
Mutually exclusive addition: $P(A \text{ or } B) = P(A) + P(B)$
Complementary rule: $P(\text{not } A) = 1 - P(A)$
Independent product rule: $P(A \text{ and } B) = P(A) \times P(B)$
Dependent events: $P(A \text{ and } B) \neq P(A) \times P(B)$

bookmarkSummary

Key Points to Remember:

The addition rule accounts for overlap by subtracting the intersection probability
Mutually exclusive events cannot occur simultaneously, so $P(A \text{ and } B) = 0$
The complementary rule is useful when it's easier to calculate the probability of an event not happening
Independence means one event doesn't affect another - test this using the product rule
Always distinguish between "with replacement" (independent) and "without replacement" (dependent) scenarios

Identities (Grade 12 NSC Matric Mathematics): Revision Notes

Identities

Understanding different types of events

The addition rule

The addition rule for mutually exclusive events

The complementary rule

The product rule for independent events

Worked examples

Key formulas summary

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