Emily, Emma, Madison, Olivia, and Hannah are five friends in middle school. On January 1st this year, they decided to each make a New Year's resolution and share it with the group. They would then meet every month and share whether they had kept or failed their resolution. (Note, once you have failed your resolution, it is failed for the entire year.)
At the beginning of the year, everyone was excited about their resolution, so each friend had a $90\%$ chance of keeping their resolution during January. (During the month they do not discuss their resolutions, so this $90\%$ is independent for all the friends.)
When they meet each month and share their progress, they are discouraged if another friend fails their resolution, making it less likely they keep their own resolution. Suppose that once a friend fails, it reduces the chance others succeed by $10\%$. For example, if $2$ friends have failed so far during the year, the chance each of the others keep their resolution during the next month is $90\%−2\times 10\%=70%$.
What is the chance that all five friends report failing their resolution during the February meeting? (Meaning they have failed in the first month.)
What is the chance that all five friends report failing their resolution during the March meeting? (Remember that once a resolution is failed it is failed for the entire year.)
Challenge: In answering the second question, you'll notice that a lot of casework is needed. For those of you with some programming experience, try to write a program to help answer/approximate the chance that all five friends report failing during the April, May, June, etc, meeting. In general try to find the probability that all five friends report failing after $12$ months. Your program can either simulate the problem (to come up with an approximate answer) or calculate the cases directly (to come up with an exact answer).
Please click here to view and participate in this weeks discussion!