Goats, Game Shows, and Grim Diagnose
May 30, 2025
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.
