Probability of two or more events

Welcome to the World of Combined Events!

In your previous studies, you probably looked at the probability of a single thing happening—like rolling a 6 on a die. But life is rarely that simple! Usually, we want to know the chances of several things happening together or one after another. For example, what are the chances of it raining and your bus being late? Or the chance of winning a game or getting a draw?

In this chapter, we are going to learn how to combine probabilities. Don't worry if this seems tricky at first; once you know whether to add or multiply, you'll find it much easier!

1. Expected Frequency: Predicting the Future

Before we look at multiple events, let’s look at how many times we expect something to happen. If you know the probability of an event, you can predict how often it will occur over a certain number of trials.

The Formula:
\( \text{Expected frequency} = n \times P(A) \)

Where:
• \( n \) is the number of times you run the experiment (the trials).
• \( P(A) \) is the probability of the event happening once.

Example: If the probability of a seed germinating is 0.7, and you plant 200 seeds, how many do you expect to grow?
\( 200 \times 0.7 = 140 \text{ seeds} \).

Quick Review: Expected frequency is just an average. It doesn't mean exactly 140 seeds will grow, but it's our best guess!

2. Mutually Exclusive Events (The "OR" Rule)

Key Term: Mutually Exclusive
Events are mutually exclusive if they cannot happen at the same time. Think of it like a light switch: it can be "On" or it can be "Off," but it cannot be both at the same moment.

The Rule:
If two events, A and B, are mutually exclusive, the probability of A or B happening is:
\( P(A \text{ or } B) = P(A) + P(B) \)

Real-World Analogy: If you are picking one fruit from a bowl containing an apple, a pear, and a banana, the events "picking an apple" and "picking a banana" are mutually exclusive. You only have one hand in the bowl!

Common Mistake: Forgetting to check if events are mutually exclusive before adding. If you are picking a card from a deck, "picking a Heart" and "picking a King" are not mutually exclusive because you could pick the King of Hearts!

Key Takeaway:

When you see the word "OR" in a probability question involving events that can't happen together, think ADDITION.

3. Independent Events (The "AND" Rule)

Key Term: Independent Events
Events are independent if the outcome of one event does not affect the outcome of the other. One event doesn't "care" what the other one did.

The Rule:
If two events, A and B, are independent, the probability of A and B happening is:
\( P(A \text{ and } B) = P(A) \times P(B) \)

Real-World Analogy: If you flip a coin and your friend in another city rolls a die, your coin land on "Heads" has zero impact on their die landing on a "6." These are independent.

Did you know? In many exam questions, we assume trials are independent unless the question tells us otherwise (like picking items from a bag without replacing them).

Key Takeaway:

When you see the word "AND" in a probability question involving independent events, think MULTIPLICATION.

4. Using Diagrams to Help

Sometimes there is too much information to keep in your head. Mathematicians use three main types of diagrams to make things clear:

1. Venn Diagrams: Great for showing overlaps between groups. If two circles don't touch, the events are mutually exclusive.

2. Tree Diagrams: Perfect for events happening one after another.
• Multiply probabilities along the branches (to find "Event A AND Event B").
• Add the results at the ends of the branches (to find "Outcome 1 OR Outcome 2").

3. Sample Space Diagrams: These are usually grids. They are brilliant for when you roll two dice or spin two spinners. You list all possible outcomes in a table so you can just count them up.

5. The "At Least One" Trick

This is a favorite topic for MEI examiners! They might ask: "What is the probability of getting at least one six in five throws of a die?"

Calculating "one six," "two sixes," "three sixes," etc., and adding them all up takes forever. Instead, use the complementary event rule.

The Logic:
The opposite of "at least one" is "none."
Since the total probability must be 1, we use this shortcut:
\( P(\text{at least one}) = 1 - P(\text{none}) \)

Step-by-Step Example: Find the probability of getting at least one 6 in 3 rolls of a die.
1. Probability of NOT getting a 6 in one roll = \( \frac{5}{6} \).
2. Probability of getting NO 6s in three rolls (Independent events) = \( \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} = \frac{125}{216} \).
3. Use the trick: \( 1 - \frac{125}{216} = \frac{91}{216} \).

Quick Review Box:

• Mutually Exclusive: Cannot happen together. Rule: \( P(A \text{ or } B) = P(A) + P(B) \).
• Independent: One doesn't affect the other. Rule: \( P(A \text{ and } B) = P(A) \times P(B) \).
• At least one: Use \( 1 - P(\text{none}) \).
• Notation: Remember that \( A' \) means "not A". \( P(A) + P(A') = 1 \).

Summary Checklist

Before moving on, make sure you can:
• Explain the difference between independent and mutually exclusive events.
• Use the multiplication rule for independent events.
• Use the addition rule for mutually exclusive events.
• Draw a tree diagram and a sample space diagram.
• Calculate the "expected frequency" using \( n \times P(A) \).
• Solve "at least one" problems using the subtraction trick.