Probability is a strange beast. Our intuition often tells us one thing, but the numbers quietly smirk and say, "Actually..."

Let’s start with something fun: goats.

The Monty Hall Problem

Imagine you're on a game show. In front of you are three doors. Behind one of them is a shiny new car. Behind the other two? Goats.

You pick a door — say, Door 1.

The host, Monty Hall, who knows what’s behind each door, opens another door — say, Door 3 — to reveal a goat. Then he turns to you and says, “Do you want to switch your choice to Door 2?”

Most people think it doesn't matter — 50/50, right?

Wrong. If you switch, your chances jump from 1/3 to 2/3. Why? Because Monty is giving you information. He always opens a goat door, so your initial 1/3 chance of picking the car is still locked in — and the remaining 2/3 probability gets lumped into the other unopened door.

Deal or No Deal: Big Monty Energy

Now imagine the same idea, but with 26 briefcases, like in Deal or No Deal. One of them has £250,000. The rest have far less.

You pick Case 7. The host then opens 24 of the remaining 25 cases — all empty or low-value. You're left with just your original case and Case 14.

Should you switch?

Absolutely.

You had a 1/26 chance of picking the jackpot at the start — so there's a 25/26 chance that it's in one of the other cases. The host just whittled those 25 down to one — essentially concentrating that 25/26 probability into a single briefcase.

Again, switching is the smart move.

Medical Statistics: When 97% Accuracy Isn’t Good Enough

Let’s switch gears to something a little darker — but equally fascinating.

Say there's a rare, fatal disease that affects 1 in a million people.

There's a test for it that's 97% accurate. Sounds reassuring, right?

Now imagine you get tested — and it comes back positive.

That sounds fucking terrifying. But what does the math say?

Let’s break it down:

• Out of 1,000,000 people:

  • 1 person actually has the disease.

  • The test has a 3% false positive rate.

  • So, 3% of 1,000,000 = 30,000 people will test positive even though they’re fine.

  • That means we now have 30,001 people who test positive — but only 1 of them is truly sick.

So if you test positive at random, your actual chance of having the disease is 1 in 30,001.

That’s incredibly low.

This is called the base rate fallacy — ignoring how rare something is when you're presented with dramatic new info. In medicine, this happens a lot.

Final Thoughts

Probability messes with your brain because it’s often at odds with your instincts.

Whether it’s choosing goats, opening briefcases, or facing a terrifying diagnosis, the truth is buried in the numbers — not in what “feels” right.

Stay sceptical. Do the maths. And always switch doors.

🧪 Try it yourself: Monty Hall Simulator