Question

1. Consider the box: [0,2,3,4,6]

a.) If 2 tickets are drawn at random ** without
replacment** from this box, what is the probability
that the sum of the draws is equal to 6?

b.) If 400 tickets are drawn at random ** with
replacment** from this box, what is the approximate
probability that the average of the draws is between 3.1 and
3.2?

Answer #1

a)

number of ways to draw 2 tickets out of 5 =5*4=20

number of ways sum is equal to 6 =5 {(0,6),(6,0),(2,4),(4,2),(3,3)}

hence probability that the sum of the draws is equal to 6 =5/20=0.25

b)

here mean value of a ticket E(X)=(0+2+3+4+6)/5=3

E(X^{2})=(0^{2}+2^{2}+3^{2}+4^{2}+6^{2})/5=13

Std deviation =sqrt(E(X^{2})-(E(X))^{2})=2

here expected mean of 400 tickets =3

and std deviiton =2/sqrt(400)=0.1

from normal approximation

probability that the average of the draws is between 3.1 and 3.2 =P(3.1<X<3.2)

=P((3.1-3)/0.1<Z<(3.2-3)/0.1)=P(1<Z<2)=0.9772-0.8413=0.1359

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b) What is the standard error for the sum of the draws?
c) What is the probability that the average of
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