## Problem of the Day #158: SeungIn’s ProgramAugust 24, 2011

Posted by Seungln in : potd , trackback

Because the slackers have neglected their duty of posting problems for MPOTD, SeungIn decides to take action and make a program that generates problems everyday at 4 p.m. However, since SeungIn’s programming is not perfect (and because SeungIn prefers some subjects to others), the distribution of the types of problems are uneven, and, once in a while, it will print out two problems instead of one. When Billy inspected the code, he found the distribution of the types of problems:

Algebra – $\displaystyle \frac{5}{18}$

Geometry – $\displaystyle \frac{1}{9}$

Number Theory – $\displaystyle \frac{7}{18}$

Combinatorics – $\displaystyle \frac{2}{9}$

Billy also finds that the program generates two problems with $\displaystyle \frac{1}{7}$ chance. However, Billy was not able to fix the code until SeungIn had to implement it to the website in order to make it make problems everyday at 4 p.m. Given the information above, what is the probability that, within a given three-week period, there will be $6$ or fewer Number Theory problems generated by SeungIn’s program?