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The Sock Drawer Probability Brain Teaser
A drawer contains 10 socks. Some are red and the rest are blue. Two socks are taken out at random, one after the other, without putting the first one back. The probability that both socks are red is one third. How many red socks are in the drawer?
Show the worked solution
What's the puzzle?
A drawer contains 10 socks. Some are red and the rest are blue. Two socks are taken out at random, one after the other, without putting the first one back. The probability that both socks are red is one third. How many red socks are in the drawer?
What's the answer?
6 red socks
How do you work it out?
- Let r be the number of red socks. The chance the first sock is red is r/10.
- One red sock has gone, so the chance the second is also red is (r - 1)/9.
- Multiply: r(r - 1)/90 = 1/3, so r(r - 1) = 30.
- Two consecutive whole numbers that multiply to 30 are 6 and 5, so r = 6.
- There are 6 red socks. Check: 6/10 × 5/9 = 30/90 = 1/3.
Frequently asked questions
Why does 'without replacement' matter in the sock drawer puzzle?
Without replacement means the first sock stays out of the drawer, so there is one fewer sock and one fewer red sock for the second pick. In the sock drawer puzzle, that changes the second probability from r/10 to (r - 1)/9.
Which GCSE skills does the sock drawer puzzle combine?
The sock drawer puzzle from LoveMaths Tutoring UK combines probability of combined events with forming and solving a quadratic equation, r(r - 1) = 30. Both skills appear on GCSE Maths papers.
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