Probability — Study Notes

Ask ten classmates how likely it is that a test question will be "hard" and you'll get ten different guesses. Probability replaces every one of those gut feelings with a single number everyone can check, so "hard" stops being an opinion and becomes something you can actually work out.

Turning "How Likely" Into a Number

A probability is always a number on the scale from 0 to 1: 0 means an outcome simply cannot occur, 1 means it must occur, and anything in between measures where its likelihood sits on that scale. To build the number, count how many outcomes would count as a "win" for whatever you're asked about, then count every outcome that could happen at all, and divide the first by the second. That division only gives a fair answer when every outcome has an equal chance of turning up — a shuffled pack, a fair spinner, an item pulled out without peeking — because unequal chances would let some outcomes unfairly outweigh others in the count.

Worked Example — Picking a Marble from a Jar

1. A jar holds 3 orange marbles, 2 purple marbles and 3 white marbles, and one marble is pulled out without looking. Start by finding how many marbles there are altogether: 3 + 2 + 3 = 8.
2. The question asks for orange, so count only the orange marbles: there are 3 of them.
3. Divide the count that matches by the total count: 3 ÷ 8 = ³⁄₈, so the chance of pulling an orange marble is ³⁄₈.

Don't be tempted to force ³⁄₈ into something rounder — an answer left as an "untidy" exact fraction is still full marks, and rounding it would actually lose accuracy.

That single-outcome method is only the starting point. Rolling two dice together, drawing a second item without putting the first one back, and combining chances with "or" versus "and" all need their own approach — worked step by step, alongside a listen-along audio walkthrough and a practice worksheet, in the full lesson below.