NCEA Level 3Binomial and PoissonAS 91586

Binomial and Poisson: at least, at most, and the rate

What this lesson covers

  • Binomial counts successes in a fixed number of trials; Poisson counts events in an interval.
  • A Poisson rate scales with the interval, so a rate per hour becomes a rate per day by multiplying.
Key idea

Two counting models

  • Binomial: successes in n trials.
  • Poisson: events in an interval.
  • Keep four decimal places throughout.
Worked example

Exactly three: P(X = 3)

  • X = oat milk orders in 10.
  • Formula, then calculator.
  1. X ~ Bin(10, 0.3)
  2. P(X = 3) = 10C3 × 0.3³ × 0.7⁷
  3. = 120 × 0.027 × 0.0823543
  4. = 0.2668
  5. Bin pdf: x = 3, n = 10, p = 0.3
Key idea

Translate the words first

  • At most 2: 2 or fewer.
  • Fewer than 2: 1 or fewer.
  • At least 4: one minus 3 or fewer.
WordsMeansCalculate
at most 2X ≤ 2P(X ≤ 2)
fewer than 2X ≤ 1P(X ≤ 1)
at least 4X ≥ 41 − P(X ≤ 3)
more than 4X ≥ 51 − P(X ≤ 4)
Worked example

At most two: P(X ≤ 2)

  • Add P(0), P(1) and P(2).
  • Or cumulative binomial.
  1. P(X ≤ 2) = P(0) + P(1) + P(2)
  2. P(0) = 0.7¹⁰ = 0.0282
  3. P(1) = 10 × 0.3 × 0.7⁹ = 0.1211
  4. P(2) = 45 × 0.3² × 0.7⁸ = 0.2335
  5. = 0.0282 + 0.1211 + 0.2335
  6. = 0.3828
  7. Bin cdf: x = 2, n = 10, p = 0.3
Worked example

At least four: P(X ≥ 4)

  • Four and every value above it.
  • Seven bars: too many to add.
  • Take the rest away from 1.
At least four: P(X ≥ 4)
Worked example

At least four: the working

  • Complement: 1 − the rest.
  • Cumulative up to 3.
  1. P(X ≥ 4) = 1 − P(X ≤ 3)complement
  2. Bin cdf: x = 3, n = 10, p = 0.3
  3. P(X ≤ 3) = 0.6496
  4. P(X ≥ 4) = 1 − 0.6496
  5. = 0.3504
Worked example

Which values are you keeping?

  • 1 − P(X ≤ 4) loses the 4.
  • Say which values you keep.
  1. 1 − P(X ≤ 4) = 1 − 0.8497drops X = 4
  2. = 0.1503
  3. 1 − P(X ≤ 3) = 1 − 0.6496keeps 4 to 10
  4. = 0.3504
Worked example

Poisson: exactly five cars

  • X = cars in 10 minutes.
  • Formula, then calculator.
  1. X ~ Po(3)
  2. P(X = 5) = e⁻³ × 3⁵ ÷ 5!
  3. = 0.049787 × 243 ÷ 120
  4. = 0.1008
  5. Poisson pdf: x = 5, λ = 3
Worked example

Scale the rate to the interval

  • Three times as long: λ = 9.
  • Write the new model.
  1. 10 minutes: λ = 3
  2. 30 minutes: λ = 3 × 3 = 9
  3. X ~ Po(9)
  4. P(X ≤ 5)
  5. Poisson cdf: x = 5, λ = 9
  6. P(X ≤ 5) = 0.1157
Worked example

Forgetting to scale

  • Scale the rate.
  • Never the probability.
  1. Poisson cdf: x = 5, λ = 3λ not scaled
  2. P(X ≤ 5) = 0.9161
  3. Poisson cdf: x = 5, λ = 9λ for 30 minutes
  4. P(X ≤ 5) = 0.1157
Worked example

Reading E−4 on the calculator

  • E−4 means × 10⁻⁴.
  • Never above 1.
  1. P(X = 0) = e⁻⁹ × 9⁰ ÷ 0!
  2. = e⁻⁹
  3. screen: 1.2341E−4
  4. = 1.2341 × 10⁻⁴
  5. = 0.00012341
  6. 1.2341above 1: impossible
Worked example

Working backwards to λ

  • None = 1 − at least one.
  • Take logs.
  1. X ~ Po(λ), per 15 minutes
  2. P(X ≥ 1) = 0.9
  3. P(X = 0) = 1 − 0.9 = 0.1
  4. e^(−λ) = 0.1
  5. −λ = ln 0.1
  6. λ = −ln 0.1 = 2.3026
  7. per hour: 4 × 2.3026 ≈ 9.21
Worked example

When algebra cannot isolate λ

  • Algebra cannot isolate λ.
  • Try values, or a solver.
  1. X ~ Po(λ), per week
  2. P(X ≤ 1) = P(X = 0) + P(X = 1)
  3. e^(−λ) + λe^(−λ) = 0.5
  4. e^(−λ)(1 + λ) = 0.5
  5. λ = 1.6: 0.5249too big
  6. λ = 1.7: 0.4932too small
  7. λ = 1.68: 0.4995just under
  8. solver: λ = 1.6783
Key idea

Where the marks are

  • Achieved: a probability, model named.
  • Merit: a rate from a probability, in context.
  • Excellence: multi-step, and judge the model.
Summary

What to take away

  • Translate the words, scale the rate, and keep four decimal places.
At most 2P(X ≤ 2)
At least 41 − P(X ≤ 3)
Poisson ratescale λ to the interval
Inversee^(−λ) = P(X = 0)
E−4 on screentimes 10 to the −4
Workingfour decimal places
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Write the model before each calculation.
  • Question 3: scale to 90 minutes first.
GivenFind
1n = 12, p = 0.4at most 3
2the same Xat least 6
32 an hour, over 90 minat most 2
4P(0 in a day) = 0.25λ per week
Show answer

1. X ~ Bin(12, 0.4): P(X ≤ 3) = 0.2253. 2. P(X ≥ 6) = 1 − P(X ≤ 5) = 1 − 0.6652 = 0.3348. 3. X ~ Po(3) for 90 minutes: P(X ≤ 2) = 0.4232. 4. e^(−λ) = 0.25, λ = 1.3863 a day, so 9.7041 a 7-day week.

Worked example

The answers: questions 1 and 2

  1. 1. X ~ Bin(12, 0.4)
  2. Bin cdf: x = 3, n = 12, p = 0.4
  3. P(X ≤ 3) = 0.2253
  4. 2. P(X ≥ 6) = 1 − P(X ≤ 5)
  5. Bin cdf: x = 5, n = 12, p = 0.4
  6. P(X ≤ 5) = 0.6652
  7. P(X ≥ 6) = 1 − 0.6652 = 0.3348
Worked example

The answers: questions 3 and 4

  1. 3. 90 minutes: λ = 2 × 1.5 = 3
  2. X ~ Po(3)
  3. Poisson cdf: x = 2, λ = 3
  4. P(X ≤ 2) = 0.4232
  5. 4. P(X = 0) = e^(−λ) = 0.25
  6. λ = −ln 0.25 = 1.3863per day
  7. week: 7 × 1.3863 = 9.7041

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