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# Strategic Intuition

Probability can often be counter to our intuition, and knowing the math behind it can protect you. Be careful with these!

Suppose you guess 20 times on a true/false test. What's the probability you'll get 19 or more correct?

Which has a better chance of happening?

A) Rolling a 1 on a single die, three times in a row.

B) Rolling a sum of 4 on two dice, two times in a row.

Suppose in a family with three children (assuming having a boy or girl is equally likely) you know that at least one of them is a girl. (This is a general fact you are aware of; it was not the case one child was randomly chosen and you found out their gender.) What is the probability there are at least two girls?

Here is a more serious application of the same mathematics:

Disease Z infects 1 out of every 1000 people. There's a test for Disease Z which is guaranteed to test positive for someone with the disease, but 1% of the time will test as positive for someone who doesn't have the disease.

You tested positive for Disease Z, and your doctor wants to place you on an expensive emergency treatment. What's the probability you actually have it?

You are in a room with three boxes.

You know that:

• One box contains two gold coins.
• Another box contains two silver coins.
• Another box contains one gold and one silver coin.

But you don't know which box is which. If you pick a box at random, and then pick one of the coins inside at random, and the coin is gold, what's the chance that the other coin in that box is also gold?

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