Strategist interview questions shared by candidates
Suppose you toss a fair coin 400 times. What is the probability that you get at least 220 heads? Round your answer to the nearest percent.
This is a central limit theorem question. The trick is to view each toss as a random variable that returns 1 if a head is tossed and 0 if a tail is tossed. Then each such random variable has expected value 1/2 and variance 1/4. So your Z-variable (for using the central limit theorem) will be: (220-200)/(sqrt(400*(1/4))) = 20/10 = 2 So we've reduced the question to asking what's the probability that Z takes a value bigger than 2. Recall that on the standard normal, the probability that z takes values between -2 and 2 is about 95%, so the probability that it takes values less than 2 is about 97.5% (it's actually more like 97.7% but just estimating). So the probability that we are bigger than 2 is a little less than 2.5%, which after rounding to the nearest percent gives us 2%.
I'm afraid I have to disagree. nice try though. What you say here is that you are using the case for Binomial distribution to turn into the Poisson approximation, but in general for sample >100, for that case to be good enough np should be <10, (in this case it's not). The most solid way is to write the probabilities for such small sample and sum the up... the probability that there are exactly 220 heads is .005405, for higher k, P gets even smaller, and summing them up gives .025417. Now if you round this guy to the nearest percentile, like the question suggests, it is 3%, rather than 2%.... One might say what's the difference, but when it comes to money, it can become HUGE. Also I must notice this is not and easy question, especially to be asked over the phone... Good luck!
What? With 400 tosses you are summing 400 binomial distributions. That's more than enough to use central limit theorem. I agree that you can do it directly from the binomial distribution (which you clearly did), but there is no way someone expects you to sum up those kinds of numbers in your head. So I'm afraid I'll have to disagree with your disagreement. :-P
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