Quiz 3: Second Level C Question
A florist makes a bouquet of 12 flowers using roses, tulips, orchids, and daisies. The type of the bouquet is defined by the numbers of the various types of flowers in the bouquet. The florist is equally likely to pick any given type of bouquet. This is to say that it is not the flowers that are randomly chosen, but it is the type of bouquet that is randomly chosen. What is the probability that the florist picks a bouquet with at least four roses given that the florist picks exactly one tulip?
AZobi: Feb. 7, 2015, 12:44 p.m.
For this question, is 15 choose 3 the total number of bouquets? I used the method mentioned in the book to get (12+4-1) choose (4-1)
AZobi: Feb. 7, 2015, 7:30 p.m.
Did anyone else get that there 8 bouquets with at least 4 roses in them given that there is exactly one tulip? Once again, I am not 100% sure, but this was my approach: R-Rose T-Tulip and x is the placement of the other flowers RRRRTxxxxxxx, RRRRRTxxxxxx, RRRRRRTxxxxx, and so on until you get 11 Roses and one Tulip. The number of bouquets added up to 8. I then put this number over the total number of bouquets (15 choose 3). Can someone please help me out?
Alan_Mendoza: Feb. 7, 2015, 8:20 p.m.
I'm not sure how to get the numerator yet, but isn't the denominator the # of ways to get exactly one tulip in the bouquet, given the formula for conditional probability in an equally likely probability space? Therefore, I believe, the denominator is 13 choose 11 using the rings and fingers method ( n-1 choose k, n-1 being the 11 flowers without the one tulip plus the three types of flowers without the tulips minus 1 and k being the 11 flowers left to choose).
Alan_Mendoza: Feb. 7, 2015, 8:24 p.m.
I believe the numerator (# of bouquets for A intersect B) could be found the same way by subtracting the 4 roses and 1 tulip that are for sure gonna be in the bouquet from the 12 flowers then adding the 3 types of flowers left (without tulips) which gives you n and k is the 7 flowers left to choose. So using n-1 choose k, it would be 9 choose 7. Please correct me if I'm wrong.
toan123: Feb. 7, 2015, 10:53 p.m.
To Alan_Mendoza: Hi, just to clarify a bit on how you came up with your solution for the numerator: How did you get the "9" part?
Alan_Mendoza: Feb. 7, 2015, 11:05 p.m.
toan123, I'm not sure if I'm right but my way of thinking was that we're guaranteed 5 flowers so we subtract that from the 12. So we end up with 7 flowers to choose from but we still need to add the types of flowers so we add 3 because the tulips are excluded. That gives us 10, then just subtract 1 to get 9. Then do 9 choose 7. Please correct me if I'm wrong.
aHean: Feb. 7, 2015, 11:14 p.m.
@Alan_Mendoza, that's what I did as well.
mtariveran: Feb. 8, 2015, 12:39 a.m.
@Alan. But didnt you already account for the 1 tulip being removed in the denominator
Alan_Mendoza: Feb. 8, 2015, 12:44 a.m.
mtariveran, denominator is # of bouquets with exactly 1 tulip and the numerator is the # of bouquets with at least 4 roses and exactly 1 tulip. They are different cases.
iraianne: Feb. 8, 2015, 2:24 a.m.
We did this is discussion. The way we set it up was: _ _ _ _ <--- 4 roses leave those alone _ _ _ _ _ _ _ <--- other flowers _ <---1 tulip leave this alone its a stars and bars question so you know that R +T +O+P= 12 flowers but you know that T=1 so you have 3 flowers meaning theres going to be 2 bars (1 less like the spaces between our fingers). now you just so the formula [(stars+ bars) choose bars] AKA [(7+2) choose 2] AKA (9 choose 2)
weisbart: Feb. 8, 2015, 10:45 a.m.
Recall the formula for conditional probability. For the numerator, you count the number of ways to get exactly one tulip and at least four roses. In the denominator, you count the number of ways to get exactly one tulip. You can use the rings on fingers method (or stars and bars---same thing) at this point to calculate the numbers, but how do you do it? Note that since you only care about the numbers of flowers of each type in the bouquet, it is a rings on fingers (or stars and bars) counting problem.
Dylan: Feb. 8, 2015, 5:52 p.m.
For my denominator, since it is the number of ways you can get exactly 1 tulip, I put 1*2^11. Logically it seems alright, but that ends up being a very large number. So I am unsure if its right. Any help here?
AZobi: Feb. 8, 2015, 8:38 p.m.
rrakha and I talked it over and we got (9 choose 2) / (13 choose 2). The nine comes from the 7 flowers left after four roses and one tulip was picked. It will then be 7+3-1 choose 3-1. . The 13 part of 13 choose two come from the 11 flowers left after choosing one tulip + the 3 flower types left -1.
martha2A: Feb. 8, 2015, 9:52 p.m.
I got the same thing as AZobi.
JCampos: Feb. 9, 2015, 7:28 a.m.
where does the 3 come from for 7+3-1 choose 3-1?
AZobi: Feb. 9, 2015, 11:12 a.m.
If there can only be one tulip, then there are only three types of flowers left that you can choose from the four.