International Mathematics (0607) | Probability of combined events

🎉 Probability of Combined Events: Study Notes (IGCSE 0607)

Welcome to one of the most practical and exciting areas of probability! Calculating the chance of a single event (like rolling a 6) is easy, but what happens when you combine two or more events? That’s what this chapter is all about! We are learning how to calculate probabilities when things happen together (A AND B) or when one of several possibilities occurs (A OR B).

Don't worry if this seems tricky at first—we have amazing visual tools (like tree diagrams) that make these problems simple!

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1. Understanding Combined Events and Key Terminology

The Two Big Questions: AND vs. OR

Combined events involve the probability of:

• Event A AND Event B happening (e.g., rolling a 6 on the first die AND rolling a 6 on the second die).
• Event A OR Event B happening (e.g., drawing an Ace OR drawing a King from a deck of cards).

Key Terms & Notation (Extended Focus)

• \(P(A)\): The probability of Event A occurring.
• \(P(A')\): The probability of Event A not occurring (Complementary event). Remember: \(P(A') = 1 - P(A)\).
• \(P(A \cup B)\): The probability of A OR B happening (The Union).
• \(P(A \cap B)\): The probability of A AND B happening (The Intersection).

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2. Mutually Exclusive Events (The Addition Rule)

Two events are mutually exclusive if they cannot happen at the same time.

Analogy: Think of turning left and turning right. You can't do both simultaneously!

The OR Rule for Mutually Exclusive Events

If A and B are mutually exclusive, the probability that A OR B happens is found by simply adding their individual probabilities.

Formula:

\(P(A \text{ or } B) = P(A \cup B) = P(A) + P(B)\)

Example: Mutually Exclusive

In a bag, the probability of picking a red counter is \(P(R) = 0.3\) and the probability of picking a blue counter is \(P(B) = 0.5\). What is the probability of picking a red OR a blue counter?

Solution: Since you cannot pick a counter that is both red and blue, the events are mutually exclusive.

\(P(R \cup B) = P(R) + P(B) = 0.3 + 0.5 = 0.8\)

What if Events Are NOT Mutually Exclusive? (Venn Diagrams)

Sometimes events can overlap (e.g., choosing a student who plays football AND plays basketball). These are not mutually exclusive.

When using Venn Diagrams (limited to two sets in this syllabus), the key is the overlap (\(A \cap B\)).

General Addition Rule:

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

We subtract \(P(A \cap B)\) because when you add \(P(A)\) and \(P(B)\), you count the overlapping region (the intersection) twice!

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3. Independent Events (The Multiplication Rule)

Two events are independent if the outcome of the first event does not affect the outcome of the second event.

Analogy: Rolling a dice and flipping a coin. The coin flip doesn't care what the dice roll was.

The AND Rule for Independent Events

The probability that Event A AND Event B both happen is found by multiplying their individual probabilities.

Formula:

\(P(A \text{ and } B) = P(A \cap B) = P(A) \times P(B)\)

This rule is frequently used in problems with replacement (e.g., picking a card, noting its color, putting it back, and then picking a second card).

Example: Independent Events (With Replacement)

The probability that a bus arrives on time (\(T\)) is 0.7. What is the probability that the bus is on time today AND on time tomorrow?

Solution: The bus's arrival today is independent of its arrival tomorrow.

\(P(T \text{ and } T) = P(T) \times P(T) = 0.7 \times 0.7 = 0.49\)

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4. Dependent Events (The Conditional Probability Rule)

Two events are dependent if the outcome of the first event changes the probability of the second event.

This usually occurs in problems without replacement (e.g., drawing two socks from a drawer, but you keep the first sock out).

The Multiplication Rule for Dependent Events (Extended)

For dependent events, you must use conditional probability—the probability of B happening given that A has already happened. While formal notation like \(P(B|A)\) is often used at higher levels, in IGCSE, you achieve this by simply updating the probabilities after the first event.

Formula:

\(P(A \text{ and } B) = P(A) \times P(\text{B after A})\)

Example: Dependent Events (Without Replacement)

A box contains 5 red balls and 5 blue balls (total 10 balls). Two balls are drawn without replacement. What is the probability of drawing two red balls?

Step 1: Probability of the first event (Red 1)
Total balls = 10. Red balls = 5.
\(P(\text{Red 1}) = \frac{5}{10}\)

Step 2: Probability of the second event (Red 2), given the first was red
Now there are only 9 balls left. Only 4 of them are red.
\(P(\text{Red 2 after Red 1}) = \frac{4}{9}\)

Step 3: Combine them (Multiply)
\(P(\text{Red 1 and Red 2}) = \frac{5}{10} \times \frac{4}{9} = \frac{20}{90} = \frac{2}{9}\)

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5. Tools for Calculating Combined Probability

5.1 Sample Space Diagrams

These are best used when combining two simple events where the number of possible outcomes is manageable (e.g., two dice, two coins, two spinners).

How to use it:

1. Create a table or list showing all possible pairs of outcomes (the "sample space").
2. Identify the specific outcomes that satisfy the combined event (A AND B, or A OR B).
3. Calculate the probability:
   \(P(\text{Event}) = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}}\)

Example: Rolling two fair six-sided dice. The sample space has 6 x 6 = 36 possible outcomes. If you want the probability of getting a total sum of 10, you look for the pairs (4, 6), (5, 5), (6, 4). There are 3 successful outcomes, so \(P(\text{sum 10}) = \frac{3}{36}\).

5.2 Tree Diagrams

Tree diagrams are essential for visualizing events that happen sequentially, especially when they are dependent (without replacement) or when you need to combine probabilities from multiple paths.

Step-by-Step Guide for Tree Diagrams:

1. Draw the Branches: Start with a point. Draw branches for the outcomes of the first event (e.g., Red or Blue).
2. Label Event 1 Probabilities: Write the probability of each outcome *by the side of the branch*.
3. Draw Second Event Branches: From the end of each Event 1 branch, draw new branches for the second event.
4. Label Event 2 Probabilities: Write the probabilities for Event 2 on these new branches. (Crucially, update these probabilities if the events are dependent!)
5. List Outcomes: Write the combined outcome (e.g., R, R) at the end of the branches.
6. Calculate Path Probabilities (AND): To find the probability of a sequence (e.g., R then R), multiply the probabilities along the path (the two branches).
7. Combine Path Probabilities (OR): If you are interested in several possible outcomes (e.g., R, B OR B, R), you add the final multiplied probabilities of those successful paths.

Did you know?

If you add up the probabilities of all the final outcomes (all the 'end of branch' probabilities), they should always sum to 1. This is a great way to check your work!

5.3 Venn Diagrams for Probability (Extended Review)

As noted earlier, Venn diagrams help calculate probabilities involving overlap (\(A \cap B\)) or the total coverage (\(A \cup B\)).

Using Venn Diagrams to Find Probabilities:

If a problem gives you probabilities for two events and their intersection, you can calculate the total probability of A or B using the formula:

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

This tool is often used when dealing with overlapping sets of data (e.g., students studying Math and Physics).

Example: If P(Maths) = 0.6, P(Physics) = 0.5, and P(Maths AND Physics) = 0.2, then:

\(P(\text{Maths OR Physics}) = 0.6 + 0.5 - 0.2 = 0.9\)

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✨ Key Takeaways for Combined Probability

1. Determine the Relationship:

• Can they happen together? (If No, Mutually Exclusive)
• Does one event change the other? (If Yes, Dependent; If No, Independent)

2. Choose the Operation:

• If calculating A OR B (Mutually Exclusive), you ADD.
• If calculating A AND B (Independent or Dependent), you MULTIPLY along the sequence.

3. Use Visual Aids:

• Use Sample Space Diagrams for two simultaneous, simple events.
• Use Tree Diagrams for sequential events, especially dependent ones (where probabilities change).
• Use Venn Diagrams when dealing with overlaps.