PUZZLE
# 012
June 13, 2016
PUZZLE
# 012
The Game: You are playing a game where you hold
your breath and flip a coin at random. You are only
allowed to breathe again when a certain sequence
of coin tosses comes up.
BREATHHOLDING
COIN TOSS
The Question: Assuming you want to breathe again
as soon as possible, should you be waiting for a
heads-heads, or heads-tails sequence of tosses to
come up (Fig. 1)?
Heads
Heads
METHOD
How We Can Look At The Problem:
Heads
Tails
This problem is about the sequence of events as a
whole. Each coin toss has a 50% chance of being
heads, and a 50% chance of being tails (Fig. 2).
Fig 1: Desired outcomes
H-H
25%
This means that instead of focusing on any single
flip, it is the overall series of coin tosses that must be
examined.
Flip 1: 50% Heads
PROCESS
1
2
Flip 2: 50% Tails
What We Know:
Both heads-heads, and heads-tails have a 25%
chance of being flipped (Fig. 3).
We can look at the average number of flips it takes
for one of these combinations to occur.
H-T
25%
Fig 2: Equal probability for each case
What Are The Limits:
An average person can hold their breath for roughly 2
minutes.
An average coin flip takes 2 seconds. This limits the
number of flips in our series to 60.
What We Find:
3
Flip 2: 50% Heads
We can get an idea of how this system works by
looking at the first 3 flips of a random sequence (Fig. 4
& 5).
Tails
Tails
Tails
Heads
Heads
Tails
Heads
Heads
Fig 3: All possible outcomes after 2 flips
The first flip is a tails (Fig. 5), which does not concern
this scenario since both winning options begin with
a heads. After the first flip both heads-heads and
heads-tails still have a 25% probability.
HT: 1
The next flip, being heads, sets up a possible winning
case. Both cases now have a 50% chance of arising.
HT: 2
Now we can look at the 2 possible scenarios from the
perspective of the losing sequence.
4
HH: 1
HT: 3
HH: 2
HT: 4
Case 1 - Tails:
If tails is flipped, the sequence is reset for heads-heads
to arise. Heads-heads once again has a 25% chance
of being flipped after 3 tosses. It will also require a
minimum of 2 more flips to be achieved (Fig. 5).
HH: 3
HT: 5
Case 2 - Heads:
5
SOLUTION
Unlike in case 1, where the probability of winning
drops to 25% when tails is flipped, the probability of
getting heads-tails remains at 50% when heads is
flipped on the third toss. This is because the sequence
is still set up to win on the fourth toss (Fig. 6).
It is important to notice that unless only heads are
flipped for the remaining 57 tosses (a 1/5.76 x 1017
chance of occurring) the heads-tails combination is
guaranteed to arise.
Heads-Tails: 5
Heads-Heads: 3
Fig 4: A sample series of 20 random flips
Flip 1 - Tails
25% chance
for both cases
Flip 2 - Heads
50% chance
for both cases
Flip 3 - Tails
25% chance
for H-H
Fig 5: Case 1: When heads-tails arises
You should be waiting for heads-tails to be flipped,
since this is likely to arise within a sequence of 4 tosses,
while heads-heads averages 6 tosses.
This is because when a winning situation is set up
for heads-heads, but a tails is flipped instead, the
probability reverts to 25%. For a heads-tails sequence,
the probability stays at 50% since the winning case
remains set up.
Flip 1 - Tails
25% chance
for both cases
Flip 2 - Heads
50% chance
for both cases
Flip 3 - Tails
50% chance
for H-T
Fig 6: Case 2: When heads-heads arises