Question 1161643
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
{{{(0.9s+0.1n)}}} 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
{{{(0.9s+0.1n)^8}}}
{{{"="}}}{{{0.9^8s^8+8(0.9^7s^7)(0.1n)+28(0.9^6s^6)(0.1^2n^2)+"..."+28(0.9^2s^2)(0.1^6n^6)+8(0.9s)(0.1^7s^7)+0.1^8s^8}}}
represent the probability that one or another of the many outcomes possible will happen.
The coefficient of the term
{{{28(0.9^2s^2)(0.1^6n^6)=28(0.9^2)(0.1^6)s^2n^6}}}
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(0.9^2)(0.1^6)=28*0.81*0.000001=0.00000081}}} or
{{{28(0.9^2)((10^-1)^6)=28*0.81*10^-6=(8.1*10^-1)*10^-6=8.1*10^-7}}} .
 
The possibility that Kevin will not spill coffee on himself while driving on any of those 8 days is the coefficient o the term
{{{0.1^8n^8}}} , which is {{{0.1^8=1*10^-0.1*10^-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(0.9s)(0.1^7s^7)=8*0.9*0.1^7sn^7}}} , which is
{{{8*0.9*0.1^7=7.2*10-7}}} 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*10^-7+0.1*10^-7=7.3*10^-7}}} or {{{0.00000072+0.00000001=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}}} .