## Ask Professor Puzzler

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Sarah asks, "On your site you asked the following question and i believe your answer is incorrect. "I have a drawer with 10 socks. 6 are blue and 4 are red. I draw a blue sock randomly and then I draw a red sock randomly. Are these independent or dependent events". You answered even though it is without replacement these are independent. I believe the answer should be dependent, since there is one less sock in the draw when you pick the red one. Am I correct?"

Hi Sarah, thanks for asking this question. I went back and looked at the page your question refers to (Independent and Dependent Events) and realized that the question you were asking about may be a bit ambiguous in how it's worded. Does it mean:

- I drew a sock specifically from the set of blue socks (in other words, I looked in the drawer to find the blue socks, and then randomly selected from that subset) or...
- I reached into the drawer without looking, and randomly pulled out a sock from the entire set, and that randomly selected sock happened to be blue.

I write competition math problems for various math leagues, and I always hate writing probability problems, because they can be so easily written in an ambiguous way (my proofreader hates proofreading them for the same reason). In this case, let's take a look at these two possible interpretations of the problem.

In the first case, the two events are clearly independent; it doesn't matter *which *of the blue socks was chosen; there are still 4 red socks, and the probability of choosing any particular red sock is 1/4. Thus, the second event is not affected by the first event.

In the second case, I randomly pulled a sock from the drawer, but now we're given the additional information that this sock happened to be blue. So this means that when I reach back into the drawer, there are now nine socks to choose from (not four, as in the previous case, because we assume I'm picking from the entire contents of the drawer.) Since we know that the sock I first chose is blue, there are still 4 red socks, so the probability of choosing a red is 4/9. We get a different answer if we read it this way, but we still have two events that don't affect each other. Since we know the first sock was blue, it doesn't matter *which *blue sock it was. The specifics of the draw don't affect the outcome of the second draw.

Here's how to make these two events *dependent*: don't specify that the first draw was blue. Now the result of the second draw is very much dependent on whether or not the first draw was blue. That's probably the situation you were thinking of.

Thank you for asking the question - as a result of your question, I'm going to do some tweaking in the wording of that problem. I don't want it to be ambiguous - especially since the second reading of the problem delves into conditional probabilities, which I don't address on that page!