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! 💪

Okay, so life is full of 'maybes'. Will you pass your driving test? Will your favourite artist drop a surprise album? Probability is just the cool mathematical way of putting a number on that 'maybe'. Think of it like a scale, or a slider, that goes from 0 to 1.

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

Worked example

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.

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

\text{Expected Frequency} = \text{Probability} \times \text{Number of trials}

Worked example

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

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.

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.

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.

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

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?

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