Coin Flip Probability Calculator
Calculate binomial probabilities for coin tosses with advanced statistical analysis
Binomial Probability Calculator
Total number of coin tosses
Target number of heads in the experiment
Type of probability calculation
0.5 for fair coin, adjust for biased coins
Probability Results
Formula Used
P(X = 5) = C(10, 5) × 0.5^5 × 0.500^5
Where: C(n,k) = n! / (k! × (n-k)!) = 252
Probability Distribution
| Heads (k) | Probability | Percentage | Bar |
|---|---|---|---|
| 0 | 0.000977 | 0.10% | |
| 1 | 0.009766 | 0.98% | |
| 2 | 0.043945 | 4.39% | |
| 3 | 0.117188 | 11.72% | |
| 4 | 0.205078 | 20.51% | |
| 5 | 0.246094 | 24.61% | |
| 6 | 0.205078 | 20.51% | |
| 7 | 0.117188 | 11.72% | |
| 8 | 0.043945 | 4.39% | |
| 9 | 0.009766 | 0.98% | |
| 10 | 0.000977 | 0.10% |
Interpretation
Example: 8 Heads in 10 Flips
Problem
Question: What's the probability of getting exactly 8 heads in 10 coin flips?
Given: n = 10, k = 8, p = 0.5 (fair coin)
Solution
P(X = 8) = C(10, 8) × (0.5)⁸ × (0.5)²
C(10, 8) = 10! / (8! × 2!) = 45
P(X = 8) = 45 × (1/256) × (1/4) = 45/1024
Answer: 45/1024 ≈ 4.395% or about 1 in 23
Key Concepts
Binomial Distribution
Models success/failure events with fixed probability
Combinations
Number of ways to choose k items from n items
Independence
Each flip doesn't affect the next
Common Scenarios
Fair Coin (p = 0.5)
Standard unbiased coin with equal heads/tails chance
Biased Coin (p ≠ 0.5)
Weighted coin favoring one outcome
Multiple Trials
Repeated independent experiments
Cumulative Probability
At least / At most calculations
Understanding Coin Flip Probability
Classical Probability
Coin flip probability is a fundamental concept in classical probability theory. It demonstrates how to calculate the likelihood of specific outcomes in repeated independent trials with binary results.
Binomial Distribution
- •Models n independent trials with probability p
- •Each trial has exactly two possible outcomes
- •Probability of success remains constant
- •Trials are independent of each other
Mathematical Foundation
Exact Probability
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
Probability of exactly k successes in n trials
Combination Formula
C(n,k) = n! / (k! × (n-k)!)
Number of ways to choose k items from n items
Note: For large values of n, use normal approximation with continuity correction
Applications and Examples
Quality Control
Testing product defect rates in manufacturing processes
Medical Testing
Calculating probabilities in clinical trials and diagnostics
Sports Analysis
Predicting win probabilities in tournaments