# Thread: Dry cleaner & probability

1. ## Dry cleaner & probability

 The dry cleaner just delivered seventeen freshly cleaned different pairs of socks in a basket but the light bulb went out. The dry cleaner did not match the socks before delivery and I am in a hurry, so I blindly pick single socks from the basket. Is it possible that I actually will be wearing a matching pair of socks that evening after ten picks and how many times do I have to reach into that basket minimally to be guaranteed a matching pair of socks?

2. ## Re: Dry cleaner & probability

Since there are seventeen different pairs of socks, it should be obvious that you can take seventeen socks out of the basket without getting a matched pair.

3. ## Re: Dry cleaner & probability

1. No, you have to pick at least 18 times from the basket to have a matching pair of socks.

2. No, you have to pick at least three times from the basket to have a matching pair of socks.

3. Yes, you have to pick at least two times from the basket to have a matching pair of socks. This is not very likely though.

4. Yes, you have to pick at least one time from the basket to have a matching pair of socks. This is not very likely though.

5. None of the above

what would be the best option?

4. ## Re: Dry cleaner & probability

After picking the 1st single sock, remains 33 single socks. Now you start picking a 2nd single sock, if not as 1st then put it aside and pick 3rd single sock....repeating this 10 times for single sock=2,3,....11 we have the following increasing probabilities of picking the matching single sock to 1st sock:

$\displaystyle \begin{array}{rrrr} {Sock= } & 2. & \text{ P[match 1st]= } & 0.030303 \\ \text{Sock= } & 3. & \text{ P[match 1st]= } & 0.03125 \\ \text{Sock= } & 4. & \text{ P[match 1st]= } & 0.0322581 \\ \text{Sock= } & 5. & \text{ P[match 1st]= } & 0.0333333 \\ \text{Sock= } & 6. & \text{ P[match 1st]= } & 0.0344828 \\ \text{Sock= } & 7. & \text{ P[match 1st]= } & 0.0357143 \\ \text{Sock= } & 8. & \text{ P[match 1st]= } & 0.037037 \\ \text{Sock= } & 9. & \text{ P[match 1st]= } & 0.0384615 \\ \text{Sock= } & 10. & \text{ P[match 1st]= } & 0.04 \\ \text{Sock= } & 11. & \text{ P[match 1st]= } & 0.0416667 \end{array}$

these 10 picks sums to 0.3545 probability of finding the matching single sock.