Solution: 2018 Fall Final - 11

Author: Michiel Smid

Question

You flip a fair coin 5 times; the flips are independent of each other. What is the probability that in these 5 coin flips, the coin comes up heads an odd number of times?

(a)

1/4

(b)

2/3

(c)

1/2

(d)

1/3

Solution

Answer by Observation

Each flip of a fair coin has two equally likely outcomes:

$ H $ (Heads)
$ T $ (Tails)

When flipping 5 coins:

The total number of possible outcomes is: $2^5 = 32$
Half of these outcomes will have an odd number of heads, and the other half will have an even number of heads.

Why Are They Evenly Split?

For every outcome with an odd number of heads, there is a corresponding outcome with an even number of heads by flipping one coin from heads to tails (or vice versa).
This creates a 1-to-1 correspondence between odd and even outcomes.

Thus, the probability of an odd number of heads is: $P(\text{Odd number of heads}) = \frac{1}{2}$