Solution: 2018 Winter Final - 15

Author: Michiel Smid

Question

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

Which of the following is correct?

(a)

None of the above.

(b)

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

(c)

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

(d)

All of the above.

We’ll take a slow and systematic approach to this question

Let S be the set of all possible outcomes
$ |S| = 2^4 = 16$
Let's determine $A$
The first two flips are fixed: $HT$
The last 2 flips can be either heads or tails: $ 2^2 = 4 $
$ |A| = 1 \cdot 4 = 4 $
$ Pr(A) = \frac{4}{16} = \frac{1}{4} $
Let's determine $B$
The second and third flips are fixed: $TT$
The first flip can be either heads or tails: $ 2^1 = 2 $
The last flip can be either heads or tails: $ 2^1 = 2 $
$ |B| = 2 \cdot 2 = 4 $
$ Pr(B) = \frac{4}{16} = \frac{1}{4} $
Let's determine $A \cap B$
If the first two flips are $HT$ and the second and third flips are $TT$, then it looks like this: $HTTX$
X can either be heads or tails: $ 2^1 = 2 $
$ |A \cap B| = 1 \cdot 2 = 2 $
$ Pr(A \cap B) = \frac{2}{16} = \frac{1}{8} $

Now, let’s check whether it’s independent

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

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

$ \frac{1}{8} = \frac{1}{16} $

Since the two sides are not equal, the events are not independent