A binomial distribution has a probability of success = 0.2
Calculate the probability of you having at least 1 successes in 8 trials:
Binomial probability formula
| f(k;n,p) = | n! * pkqn - k | |
| k!(n - k)! |
P(x >= 1) = 1 - P(x < 1) ΣP(x = k) where (0 <= k <= 1)
Calculate q:
q = 1 - p (q represents the probability of failure)
q = 1 - 0.2
q = 0.8
Calculate n!:
n! = 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
8! = 40320
Calculate P(x = 0)
Set x = 0 for the binomial probability formula
Calculate k!:
k! = 0!
0! = 1
Calculate (n - k)!:
(n - k)! = (8 - 0)!
(n - k)! = 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
8! = 40320
Take our pieces and calculate the binomial probability:
| P(x = 0) = | 8! * 0.200.8(8 - 0) | |
| 0!(8 - 0)! |
| P(x = 0) = | 40320 * 1 * 0.88 | |
| 1 * 40320 |
| P(x = 0) = | 40320 * 1 * 0.16777216 | |
| 40320 |
| P(x = 0) = | 6764.5734912 | |
| 40320 |
P(x = 0) = 0.1678
Calculate P(x = 1)
Set x = 0 for the binomial probability formula
Calculate k!:
k! = 1!
1! = 1
Calculate (n - k)!:
(n - k)! = (8 - 1)!
(n - k)! = 7!
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
7! = 5040
Take our pieces and calculate the binomial probability:
| P(x = 1) = | 8! * 0.210.8(8 - 1) | |
| 1!(8 - 1)! |
| P(x = 1) = | 40320 * 0.2 * 0.87 | |
| 1 * 5040 |
| P(x = 1) = | 40320 * 0.2 * 0.2097152 | |
| 5040 |
| P(x = 1) = | 1691.1433728 | |
| 5040 |
P(x = 1) = 0.3355
Calculate cumulative probability
P(x > 1) = 1 - (P(x = 0) + P(x = 1))
P(x > 1) = 1 - (0.1678 + 0.3355)
P(x > 1) = 1 - 0.5033
Excel or Google Sheets formula:
Excel or Google Sheets formula:=1-BINOMDIST(1,8,0.2,TRUE)Calculate nq to see if we can use the Normal Approximation:
Since q = 1 - p, we have n(1 - p) = 8(1 - 0.2)
nq = 8(0.8)
nq = 6.4
Calculate the mean μ (expected value)
μ = np
μ = 8 x 0.2
μ = 1.6
Calculate the variance σ2
σ2 = np(1 - p)
σ2 = 8 x 0.2 x (1 - 0.2)
σ2 = 1.6 x 0.8
σ2 = 1.28
Calculate the standard deviation σ
σ = √σ2 = √np(1 - p)
σ = √1.28
σ = 1.1314
Calculate Skewness:
| Skewness = | 1 - 2p |
| √np(1 - p) |
| Skewness = | 1 - 2(0.2) |
| √8(0.2)(1 - 0.2) |
| Skewness = | 1 - 0.4) |
| √8(0.2)(0.8) |
| Skewness = | 0.6 |
| √1.28 |
Skewness = 0.46875
Calculate Kurtosis:
| Kurtosis = | 1 - 6p(1 - p) |
| np(1 - p) |
| Kurtosis = | 1 - 6(0.2)(1 - 0.2) |
| 8(0.2)(1 - 0.2) |
| Kurtosis = | 1 - (1.2)(0.8) |
| 8(0.2)(0.8) |
| Kurtosis = | 1 - 0.96 |
| 1.28 |
| Kurtosis = | 0.04 |
| 1.28 |
Kurtosis = 0.03125
Final Answer
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What is the Answer?
How does the Binomial Distribution Calculator work?
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?
f(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 distribution
- discrete probability distribution of the number of successes in a sequence of n independent experiments, with a success or failure outcome
- continuity correction factor
- the bridge between the continuous normal distribution and the discrete binomial
- event
- a set of outcomes of an experiment to which a probability is assigned.
- factorial
- The product of an integer and all the integers below it
- mean
- A statistical measurement also known as the average
- moment
- a function are quantitative measures related to the shape of the functions graph
- probability
- the 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 Outcomes - standard deviation
- a measure of the amount of variation or dispersion of a set of values. The square root of variance
- variance
- How far a set of random numbers are spead out from the mean
Example calculations for the Binomial Distribution Calculator
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