A binomial distribution has a probability of success = 0.4
Calculate the probability of you having no more than 3 successes in 10 trials:
Binomial probability formula
| f(k;n,p) = | n! * pkqn - k | |
| k!(n - k)! |
P(x <= 3) = ΣP(x = k) where (0 <= k <= 3)
Calculate q:
q = 1 - p (q represents the probability of failure)
q = 1 - 0.4
q = 0.6
Calculate n!:
n! = 10!
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
10! = 3628800
Calculate P(x = 0)
Set x = 0 for the binomial probability formula
Calculate k!:
k! = 0!
0! = 1
Calculate (n - k)!:
(n - k)! = (10 - 0)!
(n - k)! = 10!
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
10! = 3628800
Take our pieces and calculate the binomial probability:
| P(x = 0) = | 10! * 0.400.6(10 - 0) | |
| 0!(10 - 0)! |
| P(x = 0) = | 3628800 * 1 * 0.610 | |
| 1 * 3628800 |
| P(x = 0) = | 3628800 * 1 * 0.0060466176 | |
| 3628800 |
| P(x = 0) = | 21941.96594688 | |
| 3628800 |
P(x = 0) = 0.006
Calculate P(x = 1)
Set x = 0 for the binomial probability formula
Calculate k!:
k! = 1!
1! = 1
Calculate (n - k)!:
(n - k)! = (10 - 1)!
(n - k)! = 9!
9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
9! = 362880
Take our pieces and calculate the binomial probability:
| P(x = 1) = | 10! * 0.410.6(10 - 1) | |
| 1!(10 - 1)! |
| P(x = 1) = | 3628800 * 0.4 * 0.69 | |
| 1 * 362880 |
| P(x = 1) = | 3628800 * 0.4 * 0.010077696 | |
| 362880 |
| P(x = 1) = | 14627.97729792 | |
| 362880 |
P(x = 1) = 0.0403
Calculate P(x = 2)
Set x = 0 for the binomial probability formula
Calculate k!:
k! = 2!
2! = 2 * 1
2! = 2
Calculate (n - k)!:
(n - k)! = (10 - 2)!
(n - k)! = 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
8! = 40320
Take our pieces and calculate the binomial probability:
| P(x = 2) = | 10! * 0.420.6(10 - 2) | |
| 2!(10 - 2)! |
| P(x = 2) = | 3628800 * 0.16 * 0.68 | |
| 2 * 40320 |
| P(x = 2) = | 3628800 * 0.16 * 0.01679616 | |
| 80640 |
| P(x = 2) = | 9751.98486528 | |
| 80640 |
P(x = 2) = 0.1209
Calculate P(x = 3)
Set x = 0 for the binomial probability formula
Calculate k!:
k! = 3!
3! = 3 * 2 * 1
3! = 6
Calculate (n - k)!:
(n - k)! = (10 - 3)!
(n - k)! = 7!
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
7! = 5040
Take our pieces and calculate the binomial probability:
| P(x = 3) = | 10! * 0.430.6(10 - 3) | |
| 3!(10 - 3)! |
| P(x = 3) = | 3628800 * 0.064 * 0.67 | |
| 6 * 5040 |
| P(x = 3) = | 3628800 * 0.064 * 0.0279936 | |
| 30240 |
| P(x = 3) = | 6501.32324352 | |
| 30240 |
P(x = 3) = 0.215
Calculate cumulative probability
P(x <= 3) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)
P(x <= 3) = 0.006 + 0.0403 + 0.1209 + 0.215
Excel or Google Sheets formula:
Excel or Google Sheets formula:=BINOMDIST(3,10,0.4,TRUE)Calculate nq to see if we can use the Normal Approximation:
Since q = 1 - p, we have n(1 - p) = 10(1 - 0.4)
nq = 10(0.6)
nq = 6
Calculate the mean μ (expected value)
μ = np
μ = 10 x 0.4
μ = 4
Calculate the variance σ2
σ2 = np(1 - p)
σ2 = 10 x 0.4 x (1 - 0.4)
σ2 = 4 x 0.6
σ2 = 2.4
Calculate the standard deviation σ
σ = √σ2 = √np(1 - p)
σ = √2.4
σ = 1.5492
Calculate Skewness:
| Skewness = | 1 - 2p |
| √np(1 - p) |
| Skewness = | 1 - 2(0.4) |
| √10(0.4)(1 - 0.4) |
| Skewness = | 1 - 0.8) |
| √10(0.4)(0.6) |
| Skewness = | 0.2 |
| √2.4 |
Skewness = 0.083333333333333
Calculate Kurtosis:
| Kurtosis = | 1 - 6p(1 - p) |
| np(1 - p) |
| Kurtosis = | 1 - 6(0.4)(1 - 0.4) |
| 10(0.4)(1 - 0.4) |
| Kurtosis = | 1 - (2.4)(0.6) |
| 10(0.4)(0.6) |
| Kurtosis = | 1 - 1.44 |
| 2.4 |
| Kurtosis = | -0.44 |
| 2.4 |
Kurtosis = -0.18333333333333
Final Answer
Kurtosis = -0.18333333333333
What is the Answer?
Kurtosis = -0.18333333333333
How does the Binomial Distribution Calculator work?
Free Binomial Distribution Calculator - Calculates the probability of 3 separate events that follow a binomial distribution. It calculates the probability of exactly k successes, no more than k successes, and greater than k successes as well as the mean, variance, standard deviation, skewness and kurtosis.
Also calculates the normal approximation to the binomial distribution with and without the continuity correction factor
Calculates moment number t using the moment generating function
This calculator has 4 inputs.
What 3 formulas are used for the Binomial Distribution Calculator?
q = 1 - pf(k;n,p) = n! * pkqn - k/k!(n - k)!
Z = X - np/√np(1 - p)
For more math formulas, check out our Formula Dossier
What 10 concepts are covered in the Binomial Distribution Calculator?
binomial distributiondiscrete probability distribution of the number of successes in a sequence of n independent experiments, with a success or failure outcomecontinuity correction factorthe bridge between the continuous normal distribution and the discrete binomialeventa set of outcomes of an experiment to which a probability is assigned.factorialThe product of an integer and all the integers below itmeanA statistical measurement also known as the averagemomenta function are quantitative measures related to the shape of the functions graphprobabilitythe likelihood of an event happening. This value is always between 0 and 1.P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomesstandard deviationa measure of the amount of variation or dispersion of a set of values. The square root of variancevarianceHow far a set of random numbers are spead out from the mean
Example calculations for the Binomial Distribution Calculator
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