10 bernoulli trials with a success probability of 0.45

Simulate 10 bernoulli trials with:

a success probability p = 0.45

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	0.45¹0.55^{(1 - 1)}	0.45 x 1	0.45
2	Failure	0.45⁰0.55^{(1 - 0)}	1 x 0.55	0.55
3	Failure	0.45⁰0.55^{(1 - 0)}	1 x 0.55	0.55
4	Failure	0.45⁰0.55^{(1 - 0)}	1 x 0.55	0.55
5	Success	0.45¹0.55^{(1 - 1)}	0.45 x 1	0.45
6	Success	0.45¹0.55^{(1 - 1)}	0.45 x 1	0.45
7	Success	0.45¹0.55^{(1 - 1)}	0.45 x 1	0.45
8	Success	0.45¹0.55^{(1 - 1)}	0.45 x 1	0.45
9	Failure	0.45⁰0.55^{(1 - 0)}	1 x 0.55	0.55
10	Success	0.45¹0.55^{(1 - 1)}	0.45 x 1	0.45

Compare Expected to Actual Results:

Given your success probability of 0.45:
we expect 0.45 x 10 = 4.5 successes

Our actual results were 6 successes and 4 failures

Calculate the median:

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

Since q > p, 0.55 > 0.45, then our median is 0

Calculate Variance:

Variance σ² = pq or p(1 - p)

Variance σ² = (0.45)(0.55)

Variance σ² = 0.2475

Calculate Skewness:

Skewness =	q - p
	√pq

Skewness =	0.55 - 0.45
	√(0.45)(0.55)

Skewness =	0.1
	√0.2475

Skewness =	0.1
	0.49749371855331

Skewness = 0.20100756305184

Calculate Kurtosis:

Kurtosis =	1 - 6pq
	√pq

Kurtosis =	1 - 6(0.45)(0.55)
	(0.45)(0.55)

Kurtosis =	1 - 6(0.2475)
	0.2475

Kurtosis =	1 - 1.485
	0.2475

Kurtosis =	-0.485
	0.2475

Kurtosis = -1.959595959596

Calculate Entropy:

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

Entropy = -(0.55)Ln(0.55) - 0.45Ln(0.45)

Entropy = -(0.55)(-0.59783700075562) - 0.45(-0.79850769621777)

Entropy = -(-0.32881035041559) - -0.359328463298

Entropy = -0.190671536702

Final Answer

Probability = 0.45
Median = 0
Variance = 0.2475
Skewness = 0.20100756305184
Kurtosis = -1.959595959596
Entropy = -0.190671536702