Solution: 2018 Winter Midterm - 16
Author: Michiel SmidQuestion
In the fall term of 2015, Nick took COMP 2804. Nick was always sitting in the back of the classroom
and spent most of his time eating bananas.
Nick buys 25 bananas at Alexa's Banana Emporium (ABE) and 30 bananas at Shelly's Fruit Market (SFM). Nick chooses, uniformly at random, a 15-element subset of these bananas. Define the event
Nick buys 25 bananas at Alexa's Banana Emporium (ABE) and 30 bananas at Shelly's Fruit Market (SFM). Nick chooses, uniformly at random, a 15-element subset of these bananas. Define the event
- A = "the subset chosen by Nick contains exactly 7 bananas from ABE".
(a)
$\frac{{25 \choose 8} + {30 \choose 7}}{55 \choose 15}$
(b)
$\frac{{25 \choose 8} \cdot {30 \choose 7}}{55 \choose 15}$
(c)
$\frac{{25 \choose 7} + {30 \choose 8}}{55 \choose 15}$
(d)
$\frac{{25 \choose 7} \cdot {30 \choose 8}}{55 \choose 15}$
Solution
We need to choose 7 bananas from ABE from the 25 bananas: $ \binom{25}{7} $
We need to choose 8 bananas from SFM from the 30 bananas: $ \binom{30}{8} $
All 15 bananas can be chosen from the 55 bananas: $ \binom{55}{15} $
Pr$ (A) = \frac{ \binom{25}{7} \cdot \binom{30}{8}}{\binom{55}{15}} $