Probability: "Sock Drawer" style questions.

Here is a genuine probability question I once tackled:

“In your sock drawer there are 6 blue socks, 2 red socks and 2 white socks, distributed so that the probability of picking each is equal. Without looking, you draw one at put it to one side before drawing another at random. What is the probability that the two socks will be the same colour?”

This kind of question is quite common but quite complicated so you need to know where to start. The best approach is to look at it, result by result:

Option A: you get a pair of red socks.

There are 10 socks in the drawer, and 2 are red. Therefore the probability that the first one will be red is 2/10 = 1/5. Assuming that, the probability that the second will be red is 1/9 as there will only be 1 red left among the remaining 9. So the probability that the pair will be red is the multiplication of the two: 1/5 X 1/9 = 1/45.

Option B: you get a pair of white socks.

The great news is that this is also 1/45. This we can tell from the question; there’s just as many reds as whites so, assuming they’re just as likely to come up as each other, we can say the probability of a pair of white is the same as the probability of a pair red.

Option C: you get a pair of blue socks.

Of the initial 10, 6 are blue. So, the probability that the first will be blue is 6/10 = 3/5. Assuming the first s blue, that would leave 5 blue and the other 4 non-blue, so the probability that the second will be blue is 5/(5+4) = 5/9. Therefore, the probability that both will be is the multiplication of the two: 3/5 X 5/9 = 15/45. (This simplifies to 1/3 but I’ll leave it for now…)

Finally, we need to add the different options together. That means, red plus white plus blue becomes: 1/45 + 1/45 + 15/45 = 17/45. Final answer.

Just remember when to multiply your fractions and when to add.

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