Introduction to Probability

Probability: From 'No Chance' to 'Nailed It!' 🎲

Introduction

1. Introduction

Alright, let's talk probability. It might sound like something from a dusty old textbook, but you literally use it every day. Deciding if you should risk texting in class? That's probability. Wondering about the drop rate for that legendary skin in your favorite game? Probability. It's just the math behind chance. We're going to break it down so you can predict outcomes better than a TikTok trend predictor. Let's get this! 💪

2. Understanding the Probability Scale

Think of probability as a scale from 0 to 1. If something has a probability of 0, it's impossible (like rolling a 7 on a normal dice). If it has a probability of 1, it's certain (like the sun rising tomorrow). A probability of 0.5 is an even chance, like a coin toss, and everything else, like passing your driving test first time, is somewhere in between. We can write probabilities as fractions, decimals, or percentages, but never as a ratio like 3 : 5.

A horizontal line labeled 'Probability Scale'. It starts at 0, labeled 'Impossible', has 0.5 in the middle, labeled 'Even Chance', and ends at 1, labeled 'Certain'.

P(\text{Event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}

A horizontal slider bar labelled 'Probability Scale'. At the far left is '0' with the label 'Impossible'. At the far right is '1' with the label 'Certain'. In the middle is '0.5' with the label 'Even Chance'. Examples are placed along the scale, like 'Rolling a 7 on a normal dice' at 0, and 'The sun rising tomorrow' at 1.

Worked example

Worked Example: Calculating Simple Probability

A bag contains 5 red, 3 blue, and 2 green marbles. What is the probability of picking a blue marble at random?

3. The Complement Rule: Probability of an Event Not Occurring

0.3

Worked example

Worked Example: Applying the Complement Rule

Worked Example: Using the 'Not' Rule

0.65

Worked example

Worked Example: The Not Rule with Real Counts

Your Part-Time Job Playlist 🎧

You're working at a cafe and the manager has a fixed playlist of 80 songs on shuffle. The genres are: 45 Pop songs, 20 Hip-Hop songs, and 15 Indie songs. If a song is picked at random, what is the probability that it is not a Pop song? Give your answer as a fraction in its simplest form.

4. Theoretical vs. Experimental Probability

\text{Relative Frequency} = \frac{\text{Number of times an event happens}}{\text{Total number of trials}}

A line graph showing the relative frequency of rolling a '6' on a die. The x-axis is 'Number of Rolls' (from 10 to 500) and the y-axis is 'Relative Frequency'. The line fluctuates wildly at the start but gets closer and closer to the theoretical probability line at 1/6 (approx 0.167) as the number of rolls increases.

Three words to know: a fair dice or spinner gives every outcome the same chance; a biased one does not (if you roll a dice 100 times and get way more 5s than anything else, your relative frequencies are telling you it's biased); and choosing at random means every item has the same chance of being picked.

Worked example

Worked Example: Calculating Relative Frequency

You work a part-time job at a cafe. You record the hot drinks sold one morning: 40 coffees, 25 teas, and 15 hot chocolates. What is the relative frequency of a customer ordering a tea?

6. Calculating Expected Frequency Using Relative Frequency

p

Worked example

Worked Example: Expected Frequency from Relative Frequency

Worked Example: Expected Frequency using Relative Frequency

A spinner is spun 200 times. It lands on Red 80 times. If the spinner is spun another 50 times, how many times would you expect it to land on Red?

Worked example

Worked Example: Calculating Relative and Expected Frequency

The Biased Spinner Situation 🎯

Kenji suspects his four-sided spinner is biased. He spins it 200 times and records the results: Red: 84 Blue: 46 Green: 52 Yellow: 18 a) What is the relative frequency of the spinner landing on Red? b) If Kenji spins the spinner another 50 times, how many times would you expect it to land on Red?

1
$For part (a), we need to find the relative frequency (our experimental probability) for landing on Red. We use the formula: number of successful outcomes divided by the total number of trials.$
$\text{Relative Frequency (Red)} = \frac{\text{Number of times it landed on Red}}{\text{Total number of spins}}$
2
$Now we plug in the numbers from Kenji's experiment. He spun it 200 times in total, and it landed on Red 84 of those times.$
$\text{Relative Frequency (Red)} = \frac{84}{200} = 0.42$
3
$\frac{1}{4} = 0.25$
$P(\text{Red}) \approx 0.42$
4
$For part (b), we need to calculate the expected frequency. We'll use our experimental probability from part (a) to predict the outcome for the next 50 spins. The formula is Probability \times New Number of Trials.$
$\text{Expected Frequency} = P(\text{Red}) \times \text{New number of spins}$
5
$Let's substitute our values. We use our best estimate for the probability (0.42) and multiply it by the new number of trials (50).$
$\text{Expected Frequency} = 0.42 \times 50 = 21$
6
$So, based on his experiment, we would expect Kenji's spinner to land on Red about 21 times if he spins it another 50 times. It might not be exactly 21, but that's our most educated guess!$
$\text{Expected number of Reds} = 21$

Answer

\text{Expected number of Reds} = 21

8. Probability of Combined Events

Alright, let's talk about combined events. Life isn't usually about just one thing happening, right? It's more like, what are the chances you'll ace your maths test and your favourite artist drops a surprise album on the same day? That's a combined event! It's simply the probability of two or more events happening together. To figure this stuff out, we have a few awesome visual tools. First up is the sample space diagram . Think of this like a grid that shows every single possible outcome when two things happen. For example, if you roll two dice while playing a game, the sample space diagram would show all 36 possible combos, from (1,1) all the way to (6,6). It’s a super clear way to make sure you don't miss any possibilities.

A 6x6 grid showing all 36 possible outcomes of rolling two six-sided dice.

Next, we have Venn diagrams . You've probably seen these before. They're perfect for showing overlap. Imagine one circle is 'Students who have a part-time job' and another is 'Students who have their driver's license'. The overlapping section shows the students who have both. For your exam, you'll only need to worry about two circles.

A Venn diagram with two overlapping circles, labeled 'Has a part-time job' and 'Has a driver's license', with numbers in each of the three regions.

Finally, there's the tree diagram, which is like a 'choose your own adventure' map for probability. You start at one point, and branches split off for each possible outcome of the first event. Then, from the end of each of those branches, more branches split off for the second event. We write the probabilities on the branches and the final combined outcomes at the very end.

A tree diagram for picking a snack twice from a choice of an apple (A) or a banana (B), showing the paths to AA, AB, BA, and BB.

Super important note: For IGCSE Core, we only deal with events with replacement . This means the situation resets after each event. It's like picking a song from a playlist; just because you heard it once doesn't mean it's removed. The total number of outcomes stays the same for the second event. Easy peasy! 👍

Worked example

Worked Example: Using a Tree Diagram for Combined Events

Let's See It in Action: The Outfit Picker Problem 👕👟

In your wardrobe, you have 5 T-shirts: 3 are black (B) and 2 are white (W). You randomly pick a T-shirt to wear on Monday, then put it back. On Tuesday, you randomly pick another T-shirt. What is the probability that you pick a black T-shirt on both days?

1
$First, let's figure out the probabilities for a single pick. There are 5 T-shirts in total. The probability of picking a black one is the number of black shirts over the total.$
$P(\text{Black}) = \frac{\text{Number of Black T-shirts}}{\text{Total T-shirts}} = \frac{3}{5}$
2
$Similarly, let's find the probability of picking a white T-shirt. We'll need this to draw our full tree diagram.$
$P(\text{White}) = \frac{\text{Number of White T-shirts}}{\text{Total T-shirts}} = \frac{2}{5}$
3
$Now, let's set up the tree diagram for the two days. The first set of branches represents Monday's pick. The second set of branches represents Tuesday's pick. Since you put the T-shirt back ('with replacement'), the probabilities for Tuesday are exactly the same as for Monday.$
$\text{Monday: } B \ \tfrac{3}{5}, \ W \ \tfrac{2}{5} \quad \text{then Tuesday, from each: } B \ \tfrac{3}{5}, \ W \ \tfrac{2}{5}$
4
$We want to find the probability of picking a black T-shirt on both days. This corresponds to the 'Black-Black' (BB) path on our tree diagram. To find the probability of a combined path, we multiply the probabilities along the branches.$
$P(\text{B and B}) = P(\text{B on Monday}) \times P(\text{B on Tuesday})$
5
$Let's plug in the numbers and calculate the final probability. So, the chance of you rocking a black T-shirt two days in a row is 9 out of 25.$
$P(\text{B and B}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}$

Answer

P(\text{B and B}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}

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Mathematics (0580)

Introduction to Probability

1. Introduction

2. Understanding the Probability Scale

Worked Example: Calculating Simple Probability

3. The Complement Rule: Probability of an Event Not Occurring

Worked Example: Applying the Complement Rule

Worked Example: The Not Rule with Real Counts

4. Theoretical vs. Experimental Probability

Worked Example: Calculating Relative Frequency

5. Calculating Expected Frequency from Theoretical Probability

Worked Example: Calculating Expected Frequency

6. Calculating Expected Frequency Using Relative Frequency

Worked Example: Expected Frequency from Relative Frequency

Worked Example: Calculating Relative and Expected Frequency

7. The Sum of Probabilities for All Outcomes

Worked Example: Finding a Missing Probability

8. Probability of Combined Events

Worked Example: Using a Tree Diagram for Combined Events

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