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03 // Bidirectional STEM Calculators // STATISTICSμ = n × p

Binomial Distribution Expected Value Solver (μ = np)

Across n independent Bernoulli trials each having success probability p, the Binomial distribution B(n, p) has expected value μ = n × p and variance σ² = n × p × (1 - p).

μ = n × p
Pick which value you want to calculate (or type into any field):Solving for: μ — Expected Successes (E[X])
Calculated

Mean number of successes across n trials

0.01100,000.00
unit

Total independent Bernoulli trials conducted

11,000,000.00
unit

Probability of success on each trial (0 to 1)

0.00010.9999
unit
Calculated Result · Expected Successes (E[X]) (μ)
35

Step-by-Step Calculation

01Canonical Law: μ = n × p
02Given Inputs (Base Units): n = 250, p = 0.14
03Solve for μ (Expected Successes (E[X])): μ = 35

Across n independent Bernoulli trials each having success probability p, the Binomial distribution B(n, p) has expected value μ = n × p and variance σ² = n × p × (1 - p).

Real-World Reference Values

Binomial Variance (σ²)σ² = n × p × (1 - p) = 30.10

Standard deviation σ = √(np(1-p)) = 5.486

Normal Approximation Rulenp ≥ 10 and n(1-p) ≥ 10

De Moivre-Laplace Gaussian convergence condition

Poisson Rare-Event Limitn → ∞, p → 0 with np = μ

Converges to Poisson(μ)

20-Row Comparison Table

Expected Successes (E[X]) (μ) · Number of Trials (n)

Binomial Distribution Expected Value Solver (μ = np) — Number of Trials → Expected Successes (E[X])
#Number of Trials (n)Expected Successes (E[X]) (μ)Δ %
01250 Current35 0.00%
027,250.94 1,015.13 +2800.38%
0314,500.88 2,030.12 +5700.35%
0421,750.82 3,045.11 +8600.33%
0529,000.76 4,060.11 +11500.30%
0636,250.70 5,075.10 +14400.28%
0743,500.64 6,090.09 +17300.26%
0850,750.58 7,105.08 +20200.23%
0958,000.52 8,120.07 +23100.21%
1065,250.46 9,135.06 +26000.18%
1172,500.40 10,150.06 +28900.16%
1279,750.34 11,165.05 +31800.14%
1387,000.28 12,180.04 +34700.11%
1494,250.22 13,195.03 +37600.09%
15101,500.16 14,210.02 +40500.06%
16108,750.10 15,225.01 +43400.04%
17116,000.04 16,240.01 +46300.02%
18123,249.98 17,255.00 +49199.99%
19130,499.92 18,269.99 +52099.97%
20137,749.86 19,284.98 +54999.95%
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Frequently Asked Questions

How many trials n do I need to average 50 conversions at a 2.5% conversion rate?+

Click SOLVE FOR on n (Number of Trials), set μ = 50 and p = 0.025. The required traffic is n = 2,000 trials.

How do I find the standard deviation of binomial outcomes?+

Multiply the solved mean μ by (1 - p) and take the square root: σ = √(μ × (1 - p)).

What are the requirements for a Binomial experiment?+

A fixed number of trials n, binary outcomes (success/failure), constant probability p on every trial, and mutual independence between trials.