Question

Sock hop (ExH). You have 10 pairs of socks, fi ve black and fi ve blue, but they are not paired up. Instead, they are all mixed up in a drawer. It ’ s early in the morning, and you don ’ t want to turn on the lights in your dark room. How many socks must you pull out to guarantee that you have a pair of one color? How many must you pull out to have two good pairs (each pair is the same color)? How many must you pull out to be certain you have a pair of black socks?

Answer 1

You need to pull out 6 socks to guarantee a pair of one color, 11 socks to guarantee two pairs, and 7 socks to be certain you have a pair of black socks.

Step-by-step explanation:

To guarantee that you have a pair of one color when selecting from a mix of 10 pairs of socks (5 black and 5 blue), you must consider the worst-case scenario. This occurs when you alternatively pick one of each color. Therefore, after picking 5 black and 4 blue socks, the next sock you pick will guarantee a pair. So, you need to pull out 6 socks to ensure a pair of one color (5 of one color and 1 of the other).

To have two good pairs, you would continue this pattern. After the first pair is secured with 6 pulls, you would possibly pick the alternating color for the next 4 pulls, and then any subsequent sock would complete a second pair of one color. In total, this would require pulling out 11 socks to ensure two pairs.

To be certain you have a pair of black socks, prepare for the possibility of pulling out all 5 blue socks first. After these 5, the next 2 socks you pull out will be black, giving you at least one pair of black socks. Therefore, you would need to pull out 7 socks to be certain.

Answer 2

Step-by-step explanation:

Total number of pairs of socks in a drawer = 10

Number of black pairs of socks = 5

Number of blue pairs of socks = 5

A. If you picked 2 socks [black] and [blue] 3rd pick guarantees you will have one pair of either blue or black

Number of socks you pull out to guarantee that you have a pair of one color = 3 socks

B. If you want to pick 2 good pairs and your 6 picks are worst case, so 7th pick of the socks will give you good pair of two colors.

Number of socks you pull out to have two good pairs = 7 socks

C. If you want to have a pair of black socks your worst case will be you pick all 10 blue socks so another 2 socks must be black.

Number of socks you pull out to have two black socks = 12 socks