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Monte

Counting and the binomial

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  3. Master it, not yet

Repeat a yes-or-no try n times with the same chance p: the count of yeses is binomial, mean np, variance np(1 − p). Combinations count the ways it can happen.

Why it matters

Same chance each time, n tries: how many hits?

Repeating yes-or-no events (a click, a sale, a free throw) follow it, so you can tell how surprising a result is.

  • SalesA 20% closer lands 4+ deals in 10 calls about 12% of the time.
  • QualityHow many defects in a batch of 50 should worry you?
  • SportsA 75% free-throw shooter makes exactly 6 of 10 about 15% of the time.

Tennis example

You win 60% of tiebreaks. With 3 tiebreaks in a match, P(win exactly 2) = C(3,2)(0.6)²(0.4) = 43.2%. P(win at least 1) = 1 − 0.4³ = 93.6%.

Binomial: drag n and p

y = 016.8%
y = 133.6%
y = 229.4%
y = 314.7%
y = 44.6%
y = 50.9%
y = 60.1%
y = 70.0%
y = 80.0%

Mean np = 1.6. Variance np(1 − p) = 1.28. P(at least 1) = 1 − (1 − p)ⁿ = 0.8322.

Check yourself

  1. 6 people, 6 seats: arrangements?

  2. Gold and silver from 5 runners: how many orders?

  3. Pick 2 of 5 runners for a pair, order doesn't matter?

The trap

The mistake people make most on this one.

At least one is 1 − P(none). Adding P(1) + P(2) + … by hand is slow and invites mistakes.

Formulas

6 formulas and a way to remember them.
Factorial
n! = n(n − 1)(n − 2)…1, and 0! = 1
Combinations
nCx = n! / [x!(n − x)!]
Binomial
P(Y = y) = nCy · pʸ · (1 − p)ⁿ⁻ʸ
Mean and variance
E(Y) = np, V(Y) = np(1 − p)
Hypergeometric
n draws without replacement from N items, K of them successes: P(k) = C(K, k) C(N − K, n − k) / C(N, n). Mean nK/N. Excel: HYPGEOM.DIST(k, n, K, N, FALSE)
Excel
BINOM.DIST(y, n, p, FALSE) for exactly y; TRUE for at most y
Remember it
Binomial is counting wins in a fixed number of matches, when every match is the same weighted coin.

Watch and go deeper

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