Probability-Generating function- but I am not sure

[porroadventum]

1.Marina Lobo wrote 10 novels, 5 f which occupy one volume and the other 5 occupy two volumes each. A person checks out 5 volumes from Marina Lobo's shelf at random. The library holds exactly one copy of each of her novels.
(a) Show that for any i=1,2,3,4,5 the probability that the ith single volume novel checked out is1/3.
(b) Show that any i= 1,2,3,4,5 the probability that the ith two volume novel checked out is 2/21.



3. I approached part a by saying that there are 5 single volume novels and there are 15 volumes to choose from so 5/15=1/3. I thought it was correct to approach part b in the same way by saying there are 10C2 (because there are 5x2 two volume novels and we pick 2 to make up the complete novel)/15C5 because you have to choose 5 volumes from 15 which gives 15/1001 which is not 2/21! I'm not sure how to approach this. Should I use generating functions to show this? Have I got part a wrong too and just got the correct answer by coincidence?

Any help and advice would be greatly appreciated, thank you.

 

[LCKurtz]

1.Marina Lobo wrote 10 novels, 5 f which occupy one volume and the other 5 occupy two volumes each. A person checks out 5 volumes from Marina Lobo's shelf at random. The library holds exactly one copy of each of her novels.
(a) Show that for any i=1,2,3,4,5 the probability that the ith single volume novel checked out is1/3.
(b) Show that any i= 1,2,3,4,5 the probability that the ith two volume novel checked out is 2/21.



3. I approached part a by saying that there are 5 single volume novels and there are 15 volumes to choose from so 5/15=1/3. I thought it was correct to approach part b in the same way by saying there are 10C2 (because there are 5x2 two volume novels and we pick 2 to make up the complete novel)/15C5 because you have to choose 5 volumes from 15 which gives 15/1001 which is not 2/21! I'm not sure how to approach this. Should I use generating functions to show this? Have I got part a wrong too and just got the correct answer by coincidence?

Any help and advice would be greatly appreciated, thank you.

For (a), there is 1 way to pick the ith single volume. Then there are how many ways to pick the other 4?

Similarly for (b), there is 1 way to pick the ith 2 volume set. Then how many ways to pick the other 3?

And, of course, there are C(15,5) ways to pick 5 volumes.

 

[porroadventum]

Thank you so much! That is very helpful:)

 

[porroadventum]

Now I am also striggling with the 3rd part of the question- (c) Let X be the total number of complete novels that this person checks out. View each one of the 10 novels as a Beroulli trial and use parts (a) and (b) or otherwise to show that E(X)= 45/21.

I set up the Beroulli trial P(X_i)= 1/3 if book is single volume
2/21 if book is two volume
Now do I use the linearity of expectation? I'm not sure how to proceed?

 

[Ray Vickson]

Now I am also striggling with the 3rd part of the question- (c) Let X be the total number of complete novels that this person checks out. View each one of the 10 novels as a Beroulli trial and use parts (a) and (b) or otherwise to show that E(X)= 45/21.

I set up the Beroulli trial P(X_i)= 1/3 if book is single volume
2/21 if book is two volume
Now do I use the linearity of expectation? I'm not sure how to proceed?

Let X_i = 0 if (complete) novel i is not included, and X_i = 1 if it is included. The number of complete novels is N = X_1 + X_2 + ... + X_10. Can you compute E(X_1)? E(X_6)? etc.

Note: getting the distribution of the number of complete novels would be much harder, as would be the problem of finding the variance of the number of complete novels, etc, because the X_1, X_2, ... are highly dependent. However, that does not matter when finding the simple mean.

RGV