Expected Tosses for Three Consecutive Heads

A fair coin is tossed repeatedly until three consecutive tosses are heads. What is the expected number of coin tosses?

Solution:

Let x be the expected number of tosses until 3 consecutive heads occur.

The image below shows the possible scenarios that arise when starting tossing:

• You could have a perfect HHH streak, and you would just end the game in 3 tosses.
• You could roll a tail on the first try, so you would have to start from scratch, with 1 toss already made
• You could roll a head, then a tail, after which you would have to start from scratch, with 2 tosses already made
• You could roll two heads, then a tail, after which you would have to start from scratch, with 2 tosses already made

This in turn can be converted to a mathematical equation:

Multiplying by 8, this becomes:

Further simplifying the equation, we get:

So, thus, we finally arrive at our final result:

Checking our solution via code:

When possible, it’s always a good idea to check your maths with a quick simulation.

We will use python and two very common libraries: NumPy and random. We simulate 1 million games, using a while loop inside a simple “for loop”. Our for loop allows us to get through the games, while the while checks if our condition is met. In the end, we just do a mean of our results and check how it compares with our mathematical result.

import numpy as np
import random
required_heads = 3
length_array = []
for i in range(int(1e6)):
    heads = 0
    length = 0
    while heads<required_heads:
        length += 1
        if random.choice(['H','T'])=='H':
            heads += 1
        else:
            heads = 0
    length_array.append(length)     

print(f"Average number of heads needed: {np.mean(length_array)}")

Average heads needed: 13.997503

For a more in-depth look, as well as a generalization (expected number of coin tosses needed for n consecutive heads), I highly recommend you check the original video