SOLUTION: If 3 books are picked from the shelf containing 5 novels 3 books of poems and a dictionary what is the prbability that the dictionary is selected. 2) 2 novels and 1 book of poem a

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Question 1203031: If 3 books are picked from the shelf containing 5 novels 3 books of poems and a dictionary what is the prbability that the dictionary is selected.
2) 2 novels and 1 book of poem are selected
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
If 3 books are picked from the shelf containing 5 novels, 3 books of poems and a dictionary, then
(1) what is the prbability that the dictionary is selected.
(2) 2 novels and 1 book of poem are selected
~~~~~~~~~~~~~~~~~~~~ 

(1) The number of all possible triples of books is  

        total = C%5B5%2B3%2B1%5D%5E3 = C%5B9%5D%5E3 = %289%2A8%2A7%29%2F%281%2A2%2A3%29 = 84.


    The number of all favorable triples is

        favorable = C%5B1%5D%5E1%2AC%5B5%2B3%5D%5E2 = C%5B8%5D%5E2 = %288%2A7%29%2F2 = 4*7 = 28.


    THEREFORE, the answer to question (1) is

        P = favorable%2Ftotal = 28%2F84 = 1%2F3.



(2)  Having the number of total triples just calculated in (a) as 84,
     we only need to calculate the number of favorable triples, which in this case is

        favorable = C%5B5%5D%5E2%2AC%5B3%5D%5E1 = %28%285%2A2%29%2F%281%2A2%29%29%2A3 = 5*3 = 15.


    THEREFORE, the answer to question (2) is

        P = favorable%2Ftotal = 15%2F84 = 5%2F28.

Solved.

Since in this problem the order of selected books does not matter,
we use COMBINATIONS to calculate the number of total triples and favorable triples.

On  Combinations,  see introductory lessons
    - Introduction to Combinations
    - PROOF of the formula on the number of Combinations
    - Problems on Combinations
in this site.



Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


In addition to the solution to the first problem from the other tutor, using nCr numbers, here are two other more elementary ways to solve that one.

These alternative methods only work for relatively simple problems like this one; however, they are useful methods for many similar problems.

(1) Pick the books one at a time and find the probability that the dictionary is NOT selected; then the probability that it IS selected is 1 minus that probability.

P(not selected first) = 8/9
P(not selected first and not selected second) = (8/9)(7/8) = 7/9
P(not selected first and not selected second and not selected third) = (8/9)(7/8)(6/7) = 6/9 = 2/3

P(IS selected first or second or third) = 1 - 2/3 = 1/3

(2) Pick the books one at a time and find the probability that the dictionary is selected either first or second or third.

P(picked first) = 1/9
P(not picked first; picked second) = (8/9)(1/8) = 1/9
P(not picked first or second; picked third) = (8/9)(7/8)(1/7) = 1/9

P(picked first or second or third) = 1/9 + 1/9 + 1/9 = 3/9 = 1/3

Note solving your second problem by either of these methods would be very tedious because of the many different sequences in which the 2 novels and 1 book of poems could be selected. The nCr method shown by the other tutor is far more efficient for this problem.