Question

Sam puts quarters into a piggy bank every day. the number of quarters added on any given day is equally likely to be 1,2,3,4 or 5. and is independent day to day. find an approximation to the probability it takes at least 60 days to collect $50 (4 quarters= 1 dollar)

Answer 1

hre as probability of geting  1,2,3,4 or 5 are equally likely therefore probability of one event out of 5=1/5=0.2

therefore mean E(X)=xP(x)=1*0.2+2*.2+3*0.2+4*0.2+5*0.2=3

E(X²)=x²P(x)=1²*0.2+2²*.2+3²*0.2+4²*0.2+5²*0.2=11

therefore std deviation =sqrt(E(X²)-(E(X))²)=sqrt(11-3²)=1.414

hence for 60 days expected collection =60*3=180

std deviaiton =1.414*sqrt(60)=10.9545

hence from normal approximation :

probability it takes at least 60 days to collect $50(200 quarters)=P(at most 200 quarters in 60days)

=P(X<=200)=P(Z<(200.5-180)/10.9545)=P(Z<1.87)=0.9693