Ninety pulls deep, still nothing, and the banner cheerfully insists the rate is 1 per cent. Everyone in the group chat has a theory: the game hates you, the servers are rigged, you should have pulled at 3am.
The real explanation is duller and a great deal more useful. It is a topic sitting in your Elementary Mathematics (E-Math) syllabus right now, and the arithmetic runs to about two lines. The real-life application of mathematics does not usually announce itself, and a gacha banner turns out to be a better example than most textbook questions manage. Once you can do the working yourself, the mystery drains out and something more interesting takes its place.
Why A 1% Drop Rate Feels Like A Lie
This is because a 1 per cent chance per pull does not mean one guaranteed item in a hundred pulls. Each pull carries its own independent 1 in 100 chance, and independence is precisely why a drought can stretch well past the point of feeling fair.
Here is the number that surprises people. If a single pull misses with probability 0.99, then missing a hundred pulls in a row has probability 0.99 raised to the power of 100. That works out at roughly 0.366. So about 37 out of every 100 players who pull a hundred times walk away with nothing at all. They were not singled out, and nothing was rigged against them.
Flip it around and about 63 per cent do land at least one. Both groups played the same banner at the same odds, which is the entire point.
What The E-Math Syllabus Asks You To Do
Rather less than the internet suggests. The Mathematics syllabus published in SEAB’s 2026 O-Level syllabus list covers the probability of single events, the probability of simple combined events using possibility diagrams and tree diagrams, and the addition and multiplication of probabilities for mutually exclusive and independent events. Schools set their own scheme of work, so the term you meet it in varies from one place to another. What does not vary is that nearly every question turns on two decisions: can these events happen together, and does one of them affect the other. Our secondary math tuition classes spend more time on those two decisions than on the arithmetic, because the arithmetic is almost never what costs the marks.
How To Work Out The Chance Of At Least One Drop
To work out the chance of at least one drop, find the chance of failing every single time, then subtract it from 1. This complement move is the most useful habit in the whole topic, because “at least one” is painful to count forwards and almost trivial to count backwards. Take ten pulls at 1 per cent each.
A single pull misses with probability 1 – 0.01 = 0.99. Ten misses in a row therefore come to 0.99 raised to the power of 10, which is 0.9044 to four decimal places. Subtract that from 1 and the chance of at least one drop across ten pulls is 0.0956, or about 9.6 per cent. Sanity check before moving on: ten pulls at 1 per cent each has to land a little under 10 per cent and never above it, so 9.6 per cent looks right.
Mutually Exclusive Or Independent: Which One Is It?
Two events are mutually exclusive when they cannot both happen, and you add their probabilities. Two events are independent when one happening does not change the chance of the other, and you multiply.
Say a banner offers a 1 per cent chance of a five star character and a 5 per cent chance of a four star one, and a single pull can only produce one of them. Those two outcomes cannot share a pull. The chance of getting either on that pull is therefore 0.01 + 0.05 = 0.06. Nothing gets multiplied here, because nothing is happening twice.
Now take two separate pulls instead. Neither knows the other happened, so the chance of five stars on both is 0.01 x 0.01 = 0.0001. Confusing these two rules is the commonest way to drop marks in this topic, and the fix is to ask yourself “same pull or different pulls?” before writing anything down.
Where Does A Tree Diagram Help?
A tree diagram earns its keep as soon as there are two or three stages and you need every outcome set out. Two pulls, each a hit at 0.01 or a miss at 0.99, produce four outcomes: hit then hit at 0.0001, hit then miss at 0.0099, miss then hit at another 0.0099, and miss then miss at 0.9801. Add all four and you get exactly 1, which is the check examiners are hoping you will run. From there, exactly one hit comes to 0.0099 + 0.0099 = 0.0198, and at least one hit comes to 1 – 0.9801 = 0.0199.
So, Is Anyone Ever Due A Win?
No, and this is the part people find hardest to swallow. Independent events have no memory, so your next pull sits at exactly 1 per cent whether you have missed ninety times or none at all. That is also the honest reason why chasing a rare item can quietly cost a good deal more than anyone planned for.
Work out the odds on your own banner and sketch the branches, then check that your outcomes add to 1. The exam asks for the same moves in the same order, which makes this the rare topic where messing about on your phone counts as revision.
If your household argues about odds more often than it practises them, we would be glad to help. Miracle Math is registered with the Ministry of Education and has been teaching maths in Bedok and Tampines since 2007, in classes small enough for a tutor to hear how a student is reasoning and not simply mark what they wrote. Get in touch through the contact page on our site and we will suggest a class that would fit.