A record on day three but not day two

Question

Independent values arrive in order, each drawn from the same continuous distribution (no ties). Call day $k$ a record if its value beats all earlier days.

What is the probability that day 2 is not a record but day 3 is?

Accepted Answer

Day k is a record with probability \frac{1}{k}, and a useful fact is that the record events on different days are independent. Day 2 is a record with probability \frac12, so it is not a record with probability \frac12; day 3 is a record with probability \frac13. Multiplying the independent events: P(	ext{no record on day 2, record on day 3}) = \left(1 - 	frac12ight)\cdot	frac13 = 	frac12 \cdot 	frac13 = \frac{1}{6} \approx 0.167. Takeaway. Record indicators at different times are independent with probabilities 1/k, so combined record patterns just multiply.

A record on day three but not day two

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