# Fifty-five percent of the applications received for a particular credit card a
ID: 2956633 • Letter: #
Question
# Fifty-five percent of the applications received for a particular credit card are accepted. Among the next twelve applications,a. what is the probability that all will be rejected?
b. what is the probability that all will be accepted?
c. what is the probability that exactly 4 will be accepted?
d. what is the probability that fewer than 3 will be accepted?
e. Determine the expected number and the variance of the accepted applications.
Explanation / Answer
A={accepted} P(A)=0.5 a) P(no one gets accepted)=(1-P(A))^12=(1-0.5)^12=0.5^12=2.44*10^-4 b) P(everyone gets accepted)=P(A)^12=0.5^12=2.44^10^-4 c) Use the binomial distribution. From 12 applicants, choose 4 who will get accepted -->(12 "choose" 4)(0.5)^4*(0.5)^8=0.121 (n "choose" k)=n!/(k!*(n-k)!) d) P(less than 3 will get accepted)=P(0 get accepted)+P(1 gets accepted)+P(2 get accepted) P(0 get accepted)=2.44*10^-4 (from part (a)) P(1 gets accepted): use binomial distribution=(12 "choose" 1)(0.5)(0.5^11)=0.00293 P(2 gets accepted): use binomial distribution=(12 "choose" 2)(0.5)^2*(0.5)^10=0.0161 --> P(less than 3 will get accepted)=2.44*10^-4+0.00293+0.0161=0.0193 e) E(X)=p1x1+p2x2+...+pixi, where i=12 P(3 get accepted)=(12 "choose" 3)0.5^12=0.0537 P(4 get accepted)=(12 "choose" 4)0.5^12=0.121 P(5 get accepted)=(12 "choose" 5)0.5^12=0.193 P(6 get accepted)=(12 "choose" 6)0.5^12=0.226 P(7 get accepted)=(12 "choose" 7)0.5^12=0.193 P(8 get accepted)=(12 "choose" 8)0.5^12=0.121 P(9 get accepted)=(12 "choose" 9)0.5^12=0.0537 P(10 get accepted)=(12 "choose" 10)0.5^12=0.0161 P(11 get accepted)=(12 "choose" 11)0.5^12=0.00229 P(12 get accepted)=(12 "choose" 12)0.5^12=2.44*10^-4 --> E(X)= 2.44*10^-4*0+0.00229+0.0161*2+3*0.0537+4*0.121+5*0.193+6*0.226+7*0.193+8*0.121+9*0.537+10*0.0161+11*0.00229+12*2.44*10^-4=5.99=6 Var(x)=E(x^2)-(E(x))^2= 2.44*10^-4*0+0.00229+0.0161*4+9*0.0537+16*0.121+25*0.193+36*0.226+49*0.193+64*0.121+81*0.537+100*0.0161+121*0.00229+144*2.44*10^-4-6^2 =38.9-36=2.9
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