Enter Values
Binomial Formula
P(X=k) = C(n,k) pk (1-p)n-k
n = trials |
k = successes |
p = probability |
C(n,k) = n!/(k!(n-k)!)
Probability Results
P(X = k)
0.246094
P(X ≤ k)
0.623047
P(X ≥ k)
0.623047
P(a ≤ X ≤ b)
--
Mean
5.0000
Variance
2.5000
Standard Deviation
1.5811
Formula Breakdown
P(X = k) = C(n,k) pk (1-p)n-k
Model Assumptions
- Fixed number of trials (n): The number of trials is determined in advance and does not change.
- Independent trials: The outcome of each trial does not affect others.
- Constant probability (p): The probability of success remains the same for every trial.
- Binary outcomes: Each trial results in exactly one of two outcomes (success or failure).
- No correlation: Events are assumed to be uncorrelated across trials.
Educational Purpose: This calculator is for educational purposes only. Results are based on theoretical binomial distribution assumptions. Not financial advice. Market conditions may differ from model assumptions.
Frequently Asked Questions
A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It is defined by two parameters: n (number of trials) and p (probability of success on each trial). The distribution gives the probability of observing exactly k successes out of n trials.
Use binomial distribution when you have a fixed number of independent trials, each trial has exactly two outcomes (success or failure), and the probability of success remains constant across all trials. Examples include counting defaults in a loan portfolio, number of successful trades, or heads in coin flips.
P(X ≤ k) gives the probability of observing k or fewer successes. This is the cumulative distribution function (CDF) and represents the probability that the outcome falls at or below a certain threshold. For example, if P(X ≤ 3) = 0.17, there is a 17% chance of getting 3 or fewer successes.
For a binomial distribution, mean = n*p and variance = n*p*(1-p). The variance is always less than or equal to the mean because (1-p) is at most 1. Maximum variance occurs when p = 0.5. The standard deviation is the square root of the variance and measures the spread of the distribution.
Yes, binomial distribution is commonly used in finance to model discrete events such as the number of loan defaults in a portfolio, success rates of trades, or counting specific outcomes over a fixed period. However, it assumes independence between trials, which may not always hold in financial markets where contagion effects exist.
Binomial distribution assumes fixed n, constant p, and independent trials with exactly two outcomes. In practice, probabilities may change over time, events may be correlated, and outcomes may not be strictly binary. For large n with moderate p, the normal approximation can be used when np > 5 and n(1-p) > 5.
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Disclaimer
This calculator is for educational purposes only and assumes ideal binomial distribution conditions. Actual financial outcomes involve additional factors not captured by this model. This tool should not be used for investment decisions.
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