What are Coin Flip Streaks?
When tossing a coin, we are often interested in streaks or runs, where the same result occurs consecutively. Unlike a standard coin toss problem where we only count the total heads, in streaks, the order of results matters. For instance, in the sequence HHHTH, there is a streak of 3 heads in 5 coin flips, which is distinct from HHTHH, containing a run of 2 heads.
Probability Calculation
Calculating the probability of runs in coin flips is not straightforward. The article introduces the concept and explains how to derive a formula for this probability.
How to Use the Coin Toss Streak Calculator?
To determine the probability of runs in coin flips using the provided calculator, follow these steps:
- Specify the total number of coin tosses.
- Enter the desired length of streaks.
- Indicate if you’re interested in streaks of exactly this length, at least this length, or at most this length.
The results include the exact probability (fractional form) and its approximation for up to 30 tosses. Additionally, a probability distribution plot is available for up to 100 tosses.
Examples of Runs Probability in Coin Flips
Three Coin Flips
For three coin flips, with eight possible results (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT), the longest streaks of heads are 3, 2, 1, 2, 1, 1, 1, 0, respectively. The probability distribution for the longest streak is as follows:
- Streak Length 0: Probability = 1/8
- Streak Length 1: Probability = 4/8
- Streak Length 2: Probability = 2/8
- Streak Length 3: Probability = 1/8
Streaks of at Least/At Most Some Length
For streaks of at least some length:
- Streak Length 0: Probability = 1
- Streak Length 1: Probability = 7/8
- Streak Length 2: Probability = 3/8
- Streak Length 3: Probability = 1/8
For streaks of at most some length:
- Streak Length 0: Probability = 1/8
- Streak Length 1: Probability = 5/8
- Streak Length 2: Probability = 7/8
- Streak Length 3: Probability = 1
Probability of Streaks in Coin Toss
The article then delves into the mathematical derivation of the probability of streaks in coin tosses. It introduces a recurrence relation and explains the calculation based on waiting for the first tails in the sequence.
Consecutive Heads & k-step Fibonacci Sequences
The discussion extends to the relationship between consecutive heads and k-step Fibonacci sequences, highlighting the appearance of the Fibonacci sequence in the probability calculation.
Finding the Coin Toss Streak Probability – An Example
An example is provided to illustrate how to find the probability of getting a streak of at least 3 heads in 10 coin flips. The result is approximately 50.78%.
Frequently Asked Questions (FAQ)
What is a Recurrence Relation?
A recurrence relation expresses the n-th term of a sequence as a combination of the previous terms of the sequence. Initial conditions must be defined for this relation.
How Do I Find the Probability of Streaks in Coin Toss?
To find the probability of the longest head run not exceeding k:
- Compute 2 to the power of n (total coin flips).
- Determine the n-th term of the (k+1)-step Fibonacci sequence.
- Divide the result from step 2 by that from step 1.
Probability of No Consecutive Heads in 3/10 Coin Flips
The probability of no consecutive heads in 3 coin flips is 62.5%, while in 10 coin flips, it’s around 14%.
If you toss a coin 3 times, the probability of at least 2 heads is 50%, while exactly 2 heads is 37.5%.
Probability of 1 Head in 4 Tosses
The probability of at least 1 head in 4 tosses is 93.75%.
Explore the world of classical probability with confidence and a better understanding of the underlying mathematics.