P3 · Solve probability problems involving two independent or dependent events

Alberta Grade 12 Mathematics

Probability

Choosing the right method for independent and dependent events

What you will learn

• Review how to calculate the probability of one event.
• Explain the difference between independent and dependent events.
• Use a probability tree and multiplication to find the probability that two events both occur.
• Solve contextual problems and check whether the probabilities make sense.

Review: probability of one event

• Probability is favourable outcomes divided by all equally likely outcomes.
• The probability of an event is between $0$ and $1$, inclusive.

Two events: independent or dependent?

• With replacement, an item is returned before the next draw; this often leaves the probabilities unchanged.
• Without replacement, an item is not returned; the probabilities for the next draw must be recalculated.
• Multiply along a path for events that all occur. Add probabilities for separate paths that meet the question.

Read and build a probability tree

• A second-stage probability may need to be recalculated after the first event.
• A tree organizes possible outcomes; it does not change the probability rules.

Common mistakes to avoid

• Name the event and identify what changes after the first result.
• Check that each set of branches from one split totals $1$.

Worked example

Independent draws with replacement

1. Identify the event probabilities
   There are $10$ tokens, of which $4$ are red. Returning the first token means the bag is restored before the second draw. The probability of red is therefore $4/10$ on both draws, so the events are independent.
   $$P(R_1)=P(R_2)=\frac{4}{10}$$
2. Multiply along the both-red path
   The question asks for the first draw to be red and the second draw to be red. These outcomes lie on one path, so multiply the two branch probabilities.
   $$\frac{4}{10}\times\frac{4}{10}=\frac{4}{25}$$

Worked example

Dependent draws without replacement

1. Find the first-draw probability
   There are $8$ counters altogether, and $3$ are yellow. The probability that the first counter is yellow is $3/8$.
   $$P(Y_1)=\frac{3}{8}$$
2. Update for the second draw
   If the first counter is yellow, only $2$ yellow counters remain among $7$ counters. The probability of yellow on the second draw, given yellow on the first, is $2/7$. The events are dependent because the first draw changes the contents of the box.
   $$P(Y_2\mid Y_1)=\frac{2}{7}$$
3. Multiply along the both-yellow path
   Both yellow outcomes are on the same path, so multiply the first probability by the updated second probability.
   $$\frac{3}{8}\times\frac{2}{7}=\frac{3}{28}$$

Worked example

At least one alarm works

1. List the possible results
   Each alarm either sounds or does not sound. “At least one” includes three outcomes: A sounds and B sounds; A sounds and B does not; or A does not and B sounds. The only outcome that does not qualify is that neither sounds.
   $$P(\text{at least one})=1-P(\text{neither})$$
2. Find the probability that neither sounds
   Alarm A fails with probability $0.1$, and Alarm B fails with probability $0.2$. Since the alarms are independent, multiply these probabilities to find the chance that both fail.
   $$0.1\times0.2=0.02$$
3. Subtract from one
   The outcomes “at least one sounds” and “neither sounds” cover all possibilities and cannot occur together. So subtract the probability of neither sounding from $1$.
   $$1-0.02=0.98$$

Common mistakes and how to avoid them

Lesson summary

• Independent events do not change each other's probabilities. Dependent events do.
• For a sequence, multiply probabilities along a path. Use the conditional probability that matches the result already known.
• For separate paths that satisfy the question, add their probabilities.
• Check that each split totals $1$ and that the final probability is between $0$ and $1$.

Check your understanding

Key terms

Event

An outcome or group of outcomes being considered.

Independent events

Events where knowing the first occurred does not change the probability of the second.

Dependent events

Events where the first changes the probability of the second.

Conditional probability

The probability of an event when another event is known to have happened.

Probability tree

A diagram showing the possible results of events in sequence and the probability of each branch.

Continue through Math 30-2

View the complete Math 30-2 Alberta Grade 12 Mathematics curriculum and lessons

• P2 · Solve probability problems with mutually exclusive or overlapping events
• P4 · Apply the fundamental counting principle
• P1 · Interpret and assess odds and probability statements
• P6 · Count unordered selections using combinations
• LR1 · Analyze numerical and logical puzzles or games using problem-solving strategies
• LR2 · Solve problems using set notation, operations, and Venn diagrams

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation P3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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