## Discussion Forum

### New Year's Resolutions (Weekly Brain Potion Discussion)

Re: New Year's Resolutions (Weekly Brain Potion Discussion)

For all of the friends to fail during the first month, there is a 1/10^5 or a 0.001% chance. For all 5 to have failed during the second meeting, we have 6 different cases, all of which depend on the first month

Case 1:None fail. This is pretty much the same as part 1, and you get 1/100000, but you multiply by 59049/100000, as that is the probability that none fail the first month. Result:59049/10000000000

Case 2:1 fails. The probability that 1 fails the first month is 32805/100000, and the probability that the other 4 fail during the second month is 1/5^4 or 1/625. Result:6561/12500000

Case 3:2 fail.The probability that 2 fail the first month is 7290/100000, and the probability that the other 3 fail is 3/10^3 Result:19683/10000000

Case 4:3 fail.The probability that 3 fail the first month is 810/100000, and the probability that the last 2 fail is 4/10^2 Result:162/250000

Case 5:4 fail.The probability that 4 fail the first month is 45/100000, and the probability that the last fails is 1/2 Result:45/200000

Case 6:all five fail during the 1st month. We have already calculated this, as 1/100000, then as all 5 have already failed, we are left with the same number.Result:1/100000

Add all of these 6 numbers, to obtain 59049/10000000000+6561/12500000+19683/10000000+162/250000+45/200000+1/100000, which gives our result, 33820849/10000000000.

I hope I didn't make any mistakes, as there were a lot of numbers, and a ton of zeros. Hopefully I didn't add/subtract any extra zeros