Probability of Getting at Least Two Consecutive Heads in Coin Tosses

When tossing a fair coin multiple times, understanding the probability of specific outcomes is a fundamental concept in probability theory. This article explores the probability of getting at least two consecutive heads in a series of coin tosses, providing a detailed analysis and multiple perspectives on this specific problem.

Introduction

Consider a fair coin that is tossed 3 times. The objective is to determine the probability of obtaining at least two consecutive heads. This problem can be approached through a detailed examination of the possible outcomes and their probabilities.

Total Outcomes

When a fair coin is tossed 3 times, the total number of possible outcomes can be calculated as:

Total Outcomes 23 8

The 8 possible outcomes are:

HHH HHT HTH HTT THH THT TTH TTT

Outcomes with At Least Two Consecutive Heads

To identify the outcomes that contain at least two consecutive heads, we can carefully analyze each possible sequence:

HHH - 3 consecutive heads HHT - 2 consecutive heads HTH - not consecutive HTT - not consecutive THH - 2 consecutive heads THT - not consecutive TTH - not consecutive TTT - not consecutive

The outcomes that meet the condition of having at least two consecutive heads are:

HHH HHT THH

Count of Favorable Outcomes

There are a total of 3 outcomes that contain at least two consecutive heads:

HHH HHT THH

Probability Calculation

The probability (P) of getting at least two consecutive heads is calculated as follows:

P frac{text{Number of favorable outcomes}}{text{Total outcomes}} frac{3}{8}

Expected Number of Flips Before Getting HH

Now, let's explore the expected number of coin tosses required to get the sequence HH. This can be investigated using a simulation approach, such as the one demonstrated with PariGP.

{k10^6s0for i1,kp-1r-1s10t0while t1rrandom2s11if r1  p1  t1prs s1print(i, s)}

The results from the simulation show that the average number of flips required to get HH is approximately 6. This aligns with the expected mathematical result using a different approach.

Probability of Specific Outcomes

In addition to the primary problem, we can consider secondary scenarios. For example, the probability of getting exactly one head is (frac{3}{8}), and the probability of getting at least two heads is (frac{4}{8}). Since these events do not intersect, the combined probability of these events occurring (if they are independent) would be the product of their individual probabilities.

Conclusion

In conclusion, the probability of getting at least two consecutive heads when tossing a fair coin 3 times is (frac{3}{8}).

Keywords: probability, consecutive heads, coin tosses