Solution: 2018 Winter Final - 18
Author: Michiel SmidQuestion
You flip a fair red coin once, and you flip a fair blue coin once, independently of each other.
Define the random variables
and
| $X = \bigg\{$ | $1\ $ | if the red coin flip resulted in heads$,$ |
| $0\ $ | if the red coin flip resulted in tails$,$ | |
| $Y = \bigg\{$ | $1\ $ | if the blue coin flip resulted in heads$,$ |
| $0\ $ | if the blue coin flip resulted in tails$,$ |
- Z = $\min(X,Y).$
(a)
1/4
(b)
1
(c)
3/4
(d)
1/2
Solution
We can brute force the answer: $ { (0, 0), (0, 1), (1, 0), (1, 1) } $
- For $ (0,0) $, the minimum value is 0
There is a $ \frac{1}{4} $ chance of this happening - For $ (0,1) $, the minimum value is 0
There is a $ \frac{1}{4} $ chance of this happening - For $ (1,0) $, the minimum value is 0
There is a $ \frac{1}{4} $ chance of this happening - For $ (1,1) $, the minimum value is 1
There is a $ \frac{1}{4} $ chance of this happening
$ \mathbb{E}(Z) = 0 \cdot \frac{1}{4} + 0 \cdot \frac{1}{4} + 0 \cdot \frac{1}{4} + 1 \cdot \frac{1}{4} $
$ \mathbb{E}(Z) = 0 + 0 + 0 + \frac{1}{4} $
$ \mathbb{E}(Z) = \frac{1}{4} $