Independent and Dependent Events

Q: How do I prove that events are independent?

Show that $P(A \text{ and } B) = P(A) \times P(B)$.

Two events are independent if one happening does not change the probability of the other. Grade 11 introduces the product rule, a test for independence, and dependent events such as drawing without replacement.

The product rule for independent events

If $A$ and $B$ are independent:

$P (A and B) = P (A) \times P (B)$

Example. A coin is tossed and a die is rolled. The results do not affect each other:

$P (heads and a 6) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$

Combined with the addition rule: $P (A or B) = P (A) + P (B) - P (A) P (B)$ .

Testing for independence

Calculate $P (A) \times P (B)$ and compare it with $P (A and B)$ .

Equal: independent.
Not equal: dependent.

Example. $P (A) = 0, 4$ , $P (B) = 0, 25$ and $P (A and B) = 0, 1$ .

$0, 4 \times 0, 25 = 0, 1$ , so $A$ and $B$ are independent.

Independent is not the same as mutually exclusive. Mutually exclusive events with non-zero probabilities are always dependent: if one happens, the other cannot.

Dependent events without replacement

When items are drawn without replacement, the second probability depends on the first draw, because there is one fewer item.

Example. A box has 6 green and 4 yellow balls. Two are drawn.

Without replacement: $P (both green) = \frac{6}{10} \times \frac{5}{9} = \frac{1}{3}$
With replacement: $P (both green) = \frac{6}{10} \times \frac{6}{10} = 0, 36$

Conditional probability notation is not in the syllabus. Work with the changed numbers directly.

Key terms

Independent events: Events where $P (A and B) = P (A) \times P (B)$ .
Dependent events: Events where one affects the probability of the other.
Without replacement: An item is not put back before the next draw.

Common mistakes

Using the product rule for events that are not independent.
Confusing independent with mutually exclusive.
Keeping the same denominator for the second draw without replacement.
Testing independence with counts instead of probabilities.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

A coin is tossed and a die is rolled. Calculate the probability of getting heads and a 6.

Worked solution

Hidden - try the question first

Solution

The events are independent
$\frac{1}{2} \times \frac{1}{6}$
$= \frac{1}{12}$

Question 2

5 marks total

routine

$P (A) = 0, 4$ , $P (B) = 0, 25$ and $P (A and B) = 0, 1$ .

(a) Are $A$ and $B$ independent? Show calculations. (3)

(b) Calculate $P (A or B)$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) $P (A) \times P (B) = 0, 4 \times 0, 25$
$= 0, 1 = P (A and B)$
So $A$ and $B$ are independent
(b) $0, 4 + 0, 25 - 0, 1$
$= 0, 55$

Question 3

6 marks total

complex

A box contains 6 green and 4 yellow balls. Two balls are drawn one after the other.

(a) Calculate $P (both green)$ if the first ball is not replaced. (3)

(b) Calculate $P (both green)$ if the first ball is replaced. (2)

Worked solution

Hidden - try the question first

Solution

(a) $P (A) \times P (B) = 0, 4 \times 0, 25$
$= \frac{1}{3}$
(b) $\frac{6}{10} \times \frac{6}{10}$
$= 0, 36$
(c) With replacement

Question 4

8 marks total

problem-solving

$A$ and $B$ are independent events. $P (A) = x$ , $P (B) = 0, 4$ and $P (A or B) = 0, 7$ .

(a) Calculate $x$ . (4)

(b) Calculate the probability that exactly one of $A$ and $B$ occurs. (4)

Worked solution

Hidden - try the question first

Solution

(a) $P (A) \times P (B) = 0, 4 \times 0, 25$
$x + 0, 4 - 0, 4 x = 0, 7$
$0, 6 x = 0, 3$
$x = 0, 5$
(b) $P (A only) = 0, 5 - 0, 2 = 0, 3$
$P (B only) = 0, 4 - 0, 2 = 0, 2$
$0, 3 + 0, 2$
$= 0, 5$

FAQs

How do I prove that events are independent?

Show that $P (A and B) = P (A) \times P (B)$ .

Can mutually exclusive events be independent?

Not if both have non-zero probabilities. If one happens, the other cannot, so they affect each other.

Is conditional probability in the Grade 11 syllabus?

No. Use the product rule and the changed numbers for 'without replacement' questions.

The product rule for independent events

Testing for independence

Dependent events without replacement

Key terms

Common mistakes

Practice questions

FAQs

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