20 bernoulli trials with a success probability of 0.75

Simulate 20 bernoulli trials with:

a success probability p = 0.75

Bernoulli Trial Formula

p^kq^{n - k}
where p = success probability, q = 1 - p

Bernoulli Trial Table

Trial #	Success/Failure	Math Work 1	Math Work 2	Probability
1	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
2	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
3	Failure	0.75⁰0.25^{(1 - 0)}	1 x 0.25	0.25
4	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
5	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
6	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
7	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
8	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
9	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
10	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
11	Failure	0.75⁰0.25^{(1 - 0)}	1 x 0.25	0.25
12	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
13	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
14	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
15	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
16	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
17	Failure	0.75⁰0.25^{(1 - 0)}	1 x 0.25	0.25
18	Failure	0.75⁰0.25^{(1 - 0)}	1 x 0.25	0.25
19	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75
20	Success	0.75¹0.25^{(1 - 1)}	0.75 x 1	0.75

Compare Expected to Actual Results:

Given your success probability of 0.75:
we expect 0.75 x 20 = 15 successes

Our actual results were 16 successes and 4 failures

Calculate the median:

If q > p, 0
If q = p, 0.5
If q < p, 1

Since q < p, 0.25 < 0.75, then our median is 1

Calculate Variance:

Variance σ² = pq or p(1 - p)

Variance σ² = (0.75)(0.25)

Variance σ² = 0.1875

Calculate Skewness:

Skewness =	q - p
	√pq

Skewness =	0.25 - 0.75
	√(0.75)(0.25)

Skewness =	-0.5
	√0.1875

Skewness =	-0.5
	0.43301270189222

Skewness = -1.1547005383793

Calculate Kurtosis:

Kurtosis =	1 - 6pq
	√pq

Kurtosis =	1 - 6(0.75)(0.25)
	(0.75)(0.25)

Kurtosis =	1 - 6(0.1875)
	0.1875

Kurtosis =	1 - 1.125
	0.1875

Kurtosis =	-0.125
	0.1875

Kurtosis = -0.66666666666667

Calculate Entropy:

Entropy = -qLn(q) - pLn(p)

Entropy = -(0.25)Ln(0.25) - 0.75Ln(0.75)

Entropy = -(0.25)(-1.3862943611199) - 0.75(-0.28768207245178)

Entropy = -(-0.34657359027997) - -0.21576155433884

Entropy = -0.034238445661164

Final Answer

Probability = 0.75
Median = 1
Variance = 0.1875
Skewness = -1.1547005383793
Kurtosis = -0.66666666666667
Entropy = -0.034238445661164

What is the Answer?

Probability = 0.75
Median = 1
Variance = 0.1875
Skewness = -1.1547005383793
Kurtosis = -0.66666666666667
Entropy = -0.034238445661164

How does the Bernoulli Trials Calculator work?

Free Bernoulli Trials Calculator - Given a success probability p and a number of trials (n), this will simulate Bernoulli Trials and offer analysis using the Bernoulli Distribution. Also calculates the skewness, kurtosis, and entropy
This calculator has 2 inputs.

What 3 formulas are used for the Bernoulli Trials Calculator?

p^kq^{n - k}
p = success probability
q = 1 - p

What 9 concepts are covered in the Bernoulli Trials Calculator?

bernoulli trials: Repeating an experiment using a bernoulli distribution
expected value: predicted value of a variable or event
E(X) = Σx_I · P(x)
kurtosis: statistical measure describing the distribution, or skewness, of observed data around the mean. Also referred to as the volatility of volatility
mean: A statistical measurement also known as the average
median: the value separating the higher half from the lower half of a data sample,
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
skewness: measure of the asymmetry of the probability distribution of a real-valued random variable about its mean
trial: a single performance of well-defined experiment
variance: How far a set of random numbers are spead out from the mean