Solution: 2018 Winter Final - 16

Author: Michiel Smid

Question

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

Which of the following is correct?

(a)

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

(b)

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

(c)

None of the above.

(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| = 2^5 = 32$
Let's determine $A$
the first three flips are fixed: 1
the last 2 flips can be either heads or tails: $ 2^2 = 4 $
$ |A| = 1 \cdot 4 = 4 $
$ Pr(A) = \frac{4}{32} = \frac{1}{8} $
Let's determine $B$
We choose 2 of the 5 positions to be tails: $ \binom{5}{2} $
We could also choose 3 of the 5 positions to be tails: $ \binom{5}{3} $
We could also choose 4 of the 5 positions to be tails: $ \binom{5}{4} $
We could also choose 5 of the 5 positions to be tails: $ \binom{5}{5} $
$ |B| = \binom{5}{2} + \binom{5}{3} + \binom{5}{4} + \binom{5}{5} $
$ |B| = 10 + 10 + 5 + 1 $
$ |B| = 26 $
$ Pr(B) = \frac{26}{32} = \frac{13}{16} $
Let's determine $A \cap B$
If the first three flips are heads and you want to have at least two tails, then the two tails must be the last 2 flips
$ |A \cap B| = 1 $
$ Pr(A \cap B) = \frac{1}{32} $

Now, let’s check whether it’s independent

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

$ \frac{1}{32} = \frac{1}{8} \cdot \frac{13}{16} $

$ \frac{1}{32} = \frac{13}{128} $

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