## Problem of the Day #44: AP Week SicknessMay 2, 2011

Posted by Seungln in : potd , trackback

Albert is sick with a certain disease that will not be cured for at least two weeks. Conveniently enough, he started feeling ill on May 1st, the beginning of the AP week.

The AP week is divided into 20 blocks – 10 days, 2 blocks per day. The disease has a 60% chance of making Albert sick enough to not be able to come to school during a block. Albert has an 80% chance of coming to school if he’s not sick, and 30% chance of coming to school even if he’s sick.

If Albert is to take 4 non-overlapping AP tests that each take up a block, all on different days, the probability that he will miss exactly one whole day (two blocks) can be expressed by the fraction $\frac{m}{n}$. Find $n-2\cdot m$. (Assume that Albert will not skip the blocks for which he has AP tests.) (Also assume that the probability of sickness and absence is completely independent from block to block – for example, Albert can be sick but still attend school regardless one block, and then be not sick but not go to school the block immediately after that.)