Question

Cherry has n cans of orange juice and 4 cans of grape juice. She choose 2 cans of juice at random and finds that the probability of getting two cans of the same juice is less than 1/2. Suggest two possible values of n.
@ganeshie8

Answer 1

Cherry can get both orange juice or both grape juice :

probability for getting both orange juice : \(\large \mathbb{\frac{n}{n+4} *\frac{n-1}{n-1+4} } = \frac{n}{n+4} *\frac{n-1}{n+3} \)

probability for getting both grape juice : \(\large \mathbb{\frac{4}{n+4} *\frac{4-1}{n+4-1} } = \frac{4}{n+4} *\frac{3}{n+3} \)

probability of getting same juice : \(\large \mathbb{\frac{n}{n+4} *\frac{n-1}{n+3} + \frac{4}{n+4} *\frac{3}{n+3}} \)

Answer 2

see if that makes sense so far

Answer 3

yes...

Answer 4

since the probability is less than 1/2 :

\(\large \mathbb{\frac{n}{n+4} *\frac{n-1}{n+3} + \frac{4}{n+4} *\frac{3}{n+3}} < \frac{1}{2}\)

\(\large \mathbb{\frac{n(n-1) + 12}{(n+4)(n+3)} < \frac{1}{2} }\)

Answer 5

crossmultiplying and simplifying gives u :

since the probability is less than 1/2 :

\(\large \mathbb{n^2-9n + 12 < 0 }\)

Answer 6

simply pick any two values for n, that satisfy the above inequality

Answer 7

n = 2 , 3 may work ?

Answer 8

wow, inequality....i hate this. But thanks! I will do it by myself now! :)
if the answer is incorrect, i will message you. thanks a lot!

the answer is "any integral values from 2 to 7"

Answer 9

yes !
http://www.wolframalpha.com/input/?i=%28n%28n-1%29%2B12%29%2F%28%28n%2B4%29%28n%2B3%29%29+%3C+1%2F2

Answer 10

n = 2,3,4,5,6,7 will work

Answer 11

i see. thanks! After i finish this question i will go to sleep. it's 10:37 pm now. haha thanks a lot!

Answer 12

wow ! you're to the east ha... :3
have good sleep :)

Answer 13

thanks. you too. I have done it! I got 1.6277<n<7.3723
therefore, 2<n<7 , and it's correct! thanks a lot haha