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AI SL · 13.2

Probability Distributions

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13.2.5

Binomial probabilities with the calculator

13.2.5

Binomial probabilities with the calculator

Target: I can define X, state X ~ B(n, p) and find P(X = k) with the calculator's binomial pdf. (+4 more)

By the end of this section

Key points

pdf or cdf?

  • Define X, then write X ~ B(n, p) before you use the calculator.
  • Binomial pdf gives the probability of one value: \(P(X = k)\).
  • Binomial cdf gives the probability of at most k, adding from 0 up to k: \(P(X \le k)\).
  • Write the settings you used (n, p and x, or lower and upper) as part of your working.

Turning words into cdf calls

  • At most k: \(P(X \le k)\). Fewer than k: \(P(X \le k-1)\).
  • At least k: \(1 - P(X \le k-1)\). More than k: \(1 - P(X \le k)\).
  • X is a whole number, so check the end value: "fewer than 5" stops at 4, "at least 5" starts at 5.
  • A cdf can also take a lower value. P(X ≥ k) is the same as cdf with lower = k and upper = n.

The whole distribution

  • Leave x blank in the binomial pdf and the calculator lists \(P(X = 0)\) up to \(P(X = n)\). Put them in a table.
  • The probabilities in a distribution add up to 1. Use that to check your table.
  • The mean of X ~ B(n, p) is np. The next lesson gives the variance as well.

On the calculator

Binomial pdf: one value

  • Enter n, p and x.
  • Write down the settings: n, p, x.

Screenshot and video to come

Binomial cdf: a running total

  • Enter n, p, lower and upper.
  • For P(X ≤ k) use lower = 0 and upper = k.
  • For at least k, find P(X ≤ k−1) and take it from 1.

Screenshot and video to come

The whole distribution in a table

  • Put the values 0 to n in a list or a spreadsheet column.
  • Use binomial pdf with n and p and leave x as the list of values.
  • Add the column up: the total is 1.

Screenshot and video to come

Explore: the binomial distribution

n trials, each a success with probability p. X is the number of successes. Choose which values of X to add up, or press a bar.

=

E(X) = np
Var(X) = np(1 − p)
Standard deviation

Move a slider to change the distribution.

  1. Set p = 0.5. What do you notice about the shape? What happens as p moves towards 0 or towards 1?
  2. Where is the tallest bar, compared with E(X) = np?
  3. Set lower = 3 and upper = n. Which single "at most" probability would you take away from 1 to get the same answer?
  4. Keep p fixed and make n larger. What happens to the spread, and to the height of the tallest bar?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Exactly k

Each raffle ticket wins a prize with probability 0.12, independently of the other tickets. Mia buys 15 tickets. X is the number of winning tickets. Find the probability that exactly 2 tickets win.

Working
  1. Define X

    Let X be the number of winning tickets. X ~ B(15, 0.12)

    Say what X counts, then write the distribution.

  2. Binomial pdf

    From the calculator: binomial pdf with n = 15, p = 0.12, x = 2

    The pdf gives one value of X.

  3. Answer

    P(X = 2) = 0.287

    3 significant figures.

Video walkthrough coming
At least k

Each parcel a courier delivers is late with probability 0.25, independently. The courier delivers 14 parcels. X is the number of late parcels. Find the probability that at least 4 parcels are late.

Working
  1. Define X

    X ~ B(14, 0.25)

    State the distribution.

  2. Use the complement

    P(X ≥ 4) = 1 − P(X ≤ 3)

    Everything except 0, 1, 2, 3.

  3. Binomial cdf

    From the calculator: binomial cdf with n = 14, p = 0.25, lower = 0, upper = 3

    Lower 0, upper 3.

  4. Subtract

    P(X ≤ 3) = 0.5213, so P(X ≥ 4) = 1 − 0.5213 = 0.479

    The calculator can also give this directly: lower = 4, upper = 14.

Video walkthrough coming
Fewer than k

A garden centre sells packets of 9 seeds. Each seed grows into a plant with probability 0.6, independently. X is the number of seeds that grow. Find the probability that fewer than 5 seeds grow.

Working
  1. Define X

    X ~ B(9, 0.6)

    State the distribution.

  2. Whole numbers

    P(X < 5) = P(X ≤ 4)

    Fewer than 5 means 0, 1, 2, 3 or 4.

  3. Binomial cdf

    From the calculator: binomial cdf with n = 9, p = 0.6, lower = 0, upper = 4

    Lower 0, upper 4.

  4. Answer

    P(X ≤ 4) = 0.267

Video walkthrough coming
The whole distribution

Each of 6 customers independently buys a drink with probability 0.4. X is the number of customers who buy a drink. Complete the table of P(X = x) for x = 0 to 6, then give P(X = 2).

00.050.10.150.20.250.30.35−0.50.51.52.53.54.55.56.5xP(X = x)
Working
  1. Define X

    X ~ B(6, 0.4)

    n = 6, p = 0.4.

  2. List every value

    From the calculator: binomial pdf with n = 6, p = 0.4, x left blank gives P(X = 0) to P(X = 6)

    Leave x blank in the pdf and the calculator lists the whole distribution, in a list or a spreadsheet column.

  3. x0123456P(X = x)0.04670.1870.3110.2760.1380.03690.00410
    The table

    x = 0 to 6: 0.0467, 0.187, 0.311, 0.276, 0.138, 0.0369, 0.00410

    The values are in the table below the bar chart.

  4. Check

    P(X = 0) + … + P(X = 6) = 1

    A distribution must add up to 1. If it does not, look for a wrong entry.

  5. Read off

    P(X = 2) = 0.311

Video walkthrough coming
The distribution and its mean

An inspector checks 40 items. Each item is faulty with probability 0.05, independently. X is the number of faulty items. Write down the distribution of X and find the mean number of faulty items.

Working
  1. Define X

    Let X be the number of faulty items.

    Say what X counts.

  2. The distribution

    X ~ B(40, 0.05)

    n = 40 trials, p = 0.05.

  3. The mean

    E(X) = np = 40 × 0.05 = 2

    The mean of a binomial is n times p. The next lesson covers it fully.

Video walkthrough coming