Probability question, topic "Random Variables"?

Here is the question:


List the values of the given random variable X together with the probability distributions in the given scenario:





An urn contains 4 red balls and 6 white balls. Three balls are drawn with replacement. The random variable X is the number of red balls that are drawn.





I know X can only assume values of 3, 2, 1, or 0.





So I do:


1) Since there are 10 balls altogether and three balls are drawn with replacement, there are 10^3, or 1000, possible combinations for the sample size.


2) I calculate Probability (X=3) first.


a. Since there are 4^3 sequences of red balls with replacement, so the probability is 4^3/10^3=0.064.


3) I calculate P (X=2) now.


a. There will be 2 red balls, and 1 white ball.


b. There are 3 positions for the white ball, and there are 6 white balls that can go into this position.


c. There are 4^2 =16 sequences of two red balls for the remaining positions.


d. So I calculate: 3*6*16/10^3=0.228


But answer says P(X=3)=1/30 and P(X=2)=3/10.


Why??

Probability question, topic "Random Variables"?
You have the correct answers for doing the problem WITH replacement... The book has the answers for doing it WITHOUT replacement.





Here's the math for without replacement:


P(x=3) = (4/10)*(3/9)*(2/8) = 1/30


P(x=2) = (4/10)*(3/9)*(6/8)*3) = 3/10





I hope this helps!