Solution: 2018 Fall Final - 16

Author: Michiel Smid

Question

You flip a fair coin three times; these three flips are independent. Define the events

(Remember that 0 is even.) Which of the following is correct?

(a)

The events $A$ and $B$ are not independent.

(b)

None of the above.

(c)

All of the above.

(d)

The events $A$ and $B$ are independent.

Let's determine $ A $
As discussed in the solution to question 11, the probability of getting an even number of heads is the same as getting an odd number of heads
$ Pr(A) = \frac{1}{2} $
Let's determine $ B $
We choose 0 tails out of the 3 flips: $ \binom{3}{0} = 1 $
We choose 1 tail out of the 3 flips: $ \binom{3}{1} = 3 $
$ |B| = 1 + 3 = 4 $
$ Pr(B) = \frac{4}{8} = \frac{1}{2} $
Let's determine $ A \cap B $
We need an even number of heads at at most 1 tail
In the case of 0 tails, we have 3 heads, which means an odd number of heads: 0
In the case of 1 tail, we have 2 heads, which means an even number of heads
We choose 1 tail out of the 3 flips: $ \binom{3}{1} = 3 $
$ |A \cap B| = 3 $
$ Pr(A \cap B) = \frac{3}{8} $

Now, let’s check whether they’re independent

$ Pr(A \cap B) = Pr(A) \cdot Pr(B) $

$ \frac{3}{8} = \frac{1}{2} \cdot \frac{1}{2} $

$ \frac{3}{8} = \frac{1}{4} $