Interesting and irritating problem.

Question

How to deal this problem. I found this problem in math competation in 2012. But, I could not solve. Could you help me...

Uncle John has taken blood pressure drops for a long time according to the following rule: 1 drop for one day, 2 drops daily for two days, ..., 10 drops daily for ten days, 9 drops daily for nine days, ..., 2 drops daily for two days, 1 drop for one day, 2 drops daily for two days, .... One day he forgot how many drops he should take, finally he took 5 drops. What is the probability that he guessed right the daily dose? Later he remembered taking 5 drops previous day, so he calmed down that he guessed the dose correctly with high probability. What is this newer probability?

Thanks in advance.

Answer 1

Uncle John's pattern of taking blood pressure drops repeats every $98$ days, and $10$ of these days are when he takes $5$ drops. Thus, the probability that he guesses right with no other information is $10/98=5/49.$

Next, if we know that the previous day he took $5$ drops, then the day he forgot can only be one of $10$ days, the last $4$ days of each of the two cycles of $5$ drops, plus the first day of a cycle of $6$ drops and a cycle of $4$ drops. Only if it were one of the two $6$-drop days days would Uncle John guess wrong, so his probability of guessing correctly has risen to $8/10$.

Answer 2

The probability should be $\frac{2*5}{1+2+\cdots+10+\cdots+2+1}=\frac{10}{99}$, and then the new probability should be just $\frac{4}{5}.$