Solution: 2017 Fall Final - 17
Author: Michiel SmidQuestion
Let $X = \{1,2,\dots,100\}$. Let $Y$ be a uniformly random 7-element subset of $X$. Define the event
- A = "the set $Y$ contains at least one even number".
(a)
$\frac{{50 \choose 7}}{{100 \choose 7}}$
(b)
$(1/2)^{7}$
(c)
$1 - (1/2)^{7}$
(d)
$1 - \frac{{50 \choose 7}}{{100 \choose 7}}$
Solution
The size of the set is when we choose any 7 numbers from the 100 numbers: $ \binom{100}{7} $
$ |S| = \binom{100}{7} $
Let’s find the probability of the complement of A
We choose 7 odd numbers out of the 50 odd numbers: $ \binom{50}{7} $
$ Pr( \overline{A}) = \frac{ \binom{50}{7} }{ \binom{100}{7} } $
$ Pr(A) = 1 - Pr( \overline{A}) $
$ Pr(A) = 1 - \frac{ \binom{50}{7} }{ \binom{100}{7} } $