Binomial Distribution Calculator
Calculate binomial probabilities for success/failure experiments with fixed number of trials
Calculate Binomial Distribution
Total number of independent trials (1-1000)
Probability of success in each trial (0 ≤ p ≤ 1)
Required number of successes (0 ≤ r ≤ n)
Binomial Distribution Results
Formula used: P(X = r) = nCr × p^r × (1-p)^(n-r)
Where: n = 10, p = 0.5, r = 5
Combinations: 10C5 = 252
Probability Analysis
Example Calculation
Dice Game Example
Scenario: Rolling 5 dice, winning if exactly 3 show ≤4
Number of trials (n): 5 dice rolls
Number of successes (r): 3 dice showing ≤4
Probability of success (p): 4/6 = 0.667 per die
Calculation
P(X = 3) = 5C3 × 0.667³ × (1-0.667)²
P(X = 3) = 10 × 0.296 × 0.111
P(X = 3) = 0.329 = 32.9%
Binomial Distribution Properties
Fixed Trials
Number of trials (n) is fixed
Binary Outcomes
Each trial has only two outcomes
Independent
Trials are independent of each other
Constant Probability
Same probability (p) for each trial
Quick Tips
Mean = n × p
Variance = n × p × (1-p)
Standard Deviation = √variance
Use normal approximation when np ≥ 5 and n(1-p) ≥ 5
Understanding Binomial Distribution
What is Binomial Distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It's used for experiments with binary outcomes (success/failure, yes/no, heads/tails).
Common Applications
- •Quality control testing
- •Medical trials and drug effectiveness
- •Survey and polling analysis
- •Sports and game probability
Formula Components
P(X = r) = nCr × p^r × (1-p)^(n-r)
- P(X = r): Probability of exactly r successes
- n: Total number of trials
- r: Number of successes
- p: Probability of success in each trial
- nCr: Number of ways to choose r items from n
Note: All trials must be independent with constant probability of success.