Applying Laws of Probability

By the end of this lesson, learners should be able to

1. Apply the addition law for mutually exclusive events in concrete counting and selection contexts.
2. Apply the multiplication law to determine joint probabilities of independent events.
3. Solve target hitting and multi-stage sampling problems by combining the addition and multiplication laws.
4. Analyse sequential drawing with and without replacement using path multiplication and branch addition.

Key vocabulary

Addition law

Rule stating that $P(A\text{ or }B)=P(A)+P(B)$ for mutually exclusive events.

By the addition law, $P(\text{lemon or mango})=\frac{4}{15}+\frac{5}{15}=\frac{3}{5}$.

Multiplication law

Rule stating that $P(S\text{ and }T)=P(S)\times P(T)$ for independent events.

Applying the multiplication law gives $P(\text{tail and }5)=\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}$.

Compound event

An event formed by combining two or more simple events using logical operators.

Finding the probability that exactly two independent archers hit a target is a compound event.

With replacement

Sampling method where each picked item is returned before the next draw.

With replacement, drawing two balls keeps the total number of balls at $11$.

Without replacement

Sampling method where picked items are not returned, altering subsequent probabilities.

Without replacement, the second draw probability uses a reduced total of $10$ items.

Tree diagram

A branching diagram representing sequence of events and their associated probabilities.

A tree diagram displays path probabilities for drawing defective bulbs without replacement.

Lesson preview

Slide 1 · Addition and Multiplication Laws of ProbabilityOpening · 10 min

When events cannot happen together, they are mutually exclusive, so we add their probabilities using the addition law $P(A\text{ or }B)=P(A)+P(B)$. When the outcome of one event does not affect another, the events are independent, so we apply the multiplication law $P(S\text{ and }T)=P(S)\times P(T)$. For example, selecting a fruit from a basket or choosing an integer from $11$ to $30$ that is divisible by $4$ or by $9$ (no integer in that range is divisible by both) uses addition, while tossing a coin and rolling a die uses multiplication. Use Figure 1 to see how picking fruit uses addition while tossing a coin uses multiplication.

• Mutually exclusive events cannot occur simultaneously, so we add their individual probabilities together.
• The probability of picking an integer from 11 to 30 that is divisible by 4 or by 9 is found by adding, because no integer in that range is divisible by both.
• Independent events occur without affecting each other, so their joint probability uses the multiplication law.
• For independent events S and T, we compute P(S and T) = P(S) x P(T).

Worked example. Total fruits = $8+5+7=20$. Mutually exclusive fruits: $P(\text{passion})=\frac{8}{20}$, $P(\text{guava})=\frac{5}{20}$. By addition law: $P(\text{passion or guava})=\frac{8}{20}+\frac{5}{20}=\frac{13}{20}$. Coin and die trials are independent: $P(\text{tail})=\frac{1}{2}$, $P(3)=\frac{1}{6}$. By multiplication law: $P(\text{tail and }3)=\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}$. Answer: (a) $\frac{13}{20}$, (b) $\frac{1}{12}$.

Check question: A basket contains $6$ oranges, $4$ lemons, and $5$ mangoes. What is the probability of picking a lemon or a mango?

1. $\frac{4}{45}$
2. $\frac{3}{5}$
3. $\frac{2}{3}$
4. $\frac{11}{15}$

Show answer

$\frac{3}{5}$. Total fruits = $6+4+5=15$. Lemons = $4$, mangoes = $5$. Since picking a lemon and picking a mango are mutually exclusive, $P(\text{lemon or mango})=\frac{4}{15}+\frac{5}{15}=\frac{9}{15}=\frac{3}{5}$.

Slide 2 · Combining the Laws of Probability in Target ProblemsModel · 10 min

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All 12 lessons in Probability 1

1. 1Performing Probability Experiments and Finding Experimental ProbabilityPerform experiments involving probabilities in different situations
2. 2Empirical Probabilities and Experimental InvestigationsPerform experiments involving probabilities in different situations
3. 3Identifying Range of Probability and Complementary EventsIdentify the range of probability in different situations
4. 4Generating Probability Space of EventsGenerate probability space of different events
5. 5Generating Probability Space for Compound EventsGenerate probability space of different events
6. 6Determining Probability of Mutually Exclusive Events and Introduction to Independent EventsDetermine the probability of mutually exclusive events and independent events
7. 7Determining Probability of Independent Events and Multi-Stage SelectionsDetermine the probability of mutually exclusive events and independent events
8. 8Probability Bounds, Complementary Events, and the Addition LawApply the laws of probability in different situations
9. 9Applying Laws of ProbabilityApply the laws of probability in different situations
10. 10Determining Probability of Independent Events Using Tree DiagramsDetermine the probability of independent events using tree diagrams
11. 11Tree Diagrams for Multi-Stage Independent EventsDetermine the probability of independent events using tree diagrams
12. 12Applications of Probability in Real-Life ContextsAppreciate the application of probability in real-life situations