SOLUTION: Every day, there is a 90% chance that Kevin will spill coffee on himself while driving to work. Looking at the next eight days, what is the probability that Kevin will spill cof

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Question 1161643: Every day, there is a 90% chance that Kevin will spill coffee on himself while driving to work.
Looking at the next eight days, what is the probability that Kevin will spill coffee on himself on just two of those days?

Looking at the next eight days, what is the probability that Kevin will spill coffee on himself on at least two of those days?
Answer by KMST(5347) About Me  (Show Source):
You can put this solution on YOUR website!
Each day Kevin drives to work,
there is a 90% chance (p=0.9) that Kevin will spill coffee on himself while driving, and
there is a 100%-90%=10% chance (p=1-0.9=0.1) that Kevin will not spill coffee on himself while driving.
If you consider one day that Kevin drives to work, the binomial
%280.9s%2B0.1n%29 represents that p=1 possibility that one or another of the possible outcomes will happen.
The coefficient 0.9 in front of s (for spill) represents the possibility that Kevin will spill coffee on himself.
The coefficient 0.1 in front of n (for no spill) represents the possibility that Kevin will not spill coffee on himself.
considering multiple days, the possibilities multiply.
If you consider 8 days that Kevin drives to work, the power of a binomial
%280.9s%2B0.1n%29%5E8
%22=%22
represent the probability that one or another of the many outcomes possible will happen.
The coefficient of the term
28%280.9%5E2s%5E2%29%280.1%5E6n%5E6%29=28%280.9%5E2%29%280.1%5E6%29s%5E2n%5E6
represents the possibility that
Kevin will spill coffee on himself on 2 of those days,
but will not spill coffee on himself on the other 6 days.
That is
28%280.9%5E2%29%280.1%5E6%29=28%2A0.81%2A0.000001=0.00000081 or
.

The possibility that Kevin will not spill coffee on himself while driving on any of those 8 days is the coefficient o the term
0.1%5E8n%5E8 , which is 0.1%5E8=1%2A10%5E-0.1%2A10%5E-7 or 0.00000001.
The possibility that Kevin will spill coffee on himself while driving only 1 of those 8 days is the coefficient o the term
8%280.9s%29%280.1%5E7s%5E7%29=8%2A0.9%2A0.1%5E7sn%5E7 , which is
8%2A0.9%2A0.1%5E7=7.2%2A10-7 or 0.00000072 .
Adding those two possibilities,
the possibility that Kevin will sill coffe on himself on no more than 1 of those 8 days is
7.2%2A10%5E-7%2B0.1%2A10%5E-7=7.3%2A10%5E-7 or 0.00000072%2B0.00000001=0.00000073 .
The possibility of the opposite (Kevin spilling coffee on himself at least 2 of the 8 days is
1-0.00000073=0.99999927 .