Pietro MeloniPhD

IB Calculator Guide / Binomial distribution

GDC skill · binompdf · binomcdf

Binomial distribution on your calculator: pdf, cdf and "at least" questions

IB Maths AA SL and HL, AI SL and HL. Paper 2 and HL Paper 3. Expected in every session.

Every binomial question is one of three shapes: exactly kk successes (pdf), at most kk (cdf), or between two values. The calculator does all three, but the three machines disagree on whether the cumulative command takes one bound or two, and "at least" questions need a complement on the TI-84. The question almost always says "find the probability that at least ..." precisely because that is where students slip.

The example used below

A fair six-sided die is rolled 20 times. Let XX be the number of sixes. (a) Find P(X=4)P(X = 4). (b) Find P(X≤4)P(X \le 4). (c) Find P(3≤X≤7)P(3 \le X \le 7). (d) Find P(X≥5)P(X \ge 5).

Answers

X∼B(20,16)X \sim B(20, \tfrac{1}{6}). (a) 0.202. (b) 0.769. (c) 0.660. (d) 1−P(X≤4)=0.2311 - P(X \le 4) = 0.231. All to 3 s.f.

Choose your calculator

The page remembers your choice on the other guides.

TI-Nspire CX II

In one line. menu, 6 Statistics, 5 Distributions, then D Binomial Pdf or E Binomial Cdf. The Cdf takes a lower AND an upper bound, so "between" needs no subtraction.

  1. 1

    Open a Calculator page, press menu, choose Statistics, then Distributions.

    menu6: Statistics5: Distributions
  2. 2

    For exactly 4 sixes choose Binomial Pdf: Num Trials n = 20, Prob Success p = 1/6, X Value 4.

    D: Binomial Pdfn: 20p: 1/6X Value: 4OK
  3. 3

    binomPdf(20,1/6,4) = 0.20224. Write 0.202.

  4. 4

    For at most 4 choose Binomial Cdf with Lower Bound 0 and Upper Bound 4.

    E: Binomial Cdfn: 20p: 1/6Lower: 0Upper: 4OK
  5. 5

    binomCdf(20,1/6,0,4) = 0.76874. Write 0.769.

  6. 6

    For between 3 and 7 inclusive use Lower 3, Upper 7: binomCdf(20,1/6,3,7) = 0.66009. For at least 5 use Lower 5, Upper 20: 0.23126.

Typed form

  • binomPdf(n, p, k)
  • binomCdf(n, p, lower, upper)
  • binomCdf(n, p, k) with one bound means P(X ≤ k)

Where the TI-Nspire CX II trips people up

  • If you type p as a decimal, use enough digits: 0.17 instead of 1/6 changes the third significant figure. Type 1/6 directly.
  • The menu letters D and E are correct for OS 5 and 6; on an older OS the binomial entries may sit a few places higher. Look for the word Binomial, not the letter.
  • Inverse Binomial (F) answers "smallest kk such that P(X≤k)≥P(X \le k) \ge area": useful for HL Paper 3 but not for ordinary probability questions.

Casio fx-CG50

In one line. MENU, Statistics, F5 DIST, F5 BINOMIAL, then F1 Bpd or F2 Bcd. Bcd takes Lower and Upper, so "between" is direct.

  1. 1

    Open Statistics and the distribution menu.

    MENUStatisticsF5 DISTF5 BINOMIAL
  2. 2

    For exactly 4 choose Bpd. Data Variable, x 4, Numtrial 20, p 1÷6. Press EXE or F1 CALC.

    F1 BpdData: Variablex: 4Numtrial: 20p: 1÷6F1 CALC
  3. 3

    The screen shows p = 0.20224. Write 0.202.

  4. 4

    For at most 4 choose Bcd. Lower 0, Upper 4, Numtrial 20, p 1÷6.

    F2 BcdLower: 0Upper: 4Numtrial: 20p: 1÷6F1 CALC
  5. 5

    p = 0.76874. Write 0.769.

  6. 6

    For between 3 and 7 set Lower 3, Upper 7 (0.66009). For at least 5 set Lower 5, Upper 20 (0.23126).

Typed form

  • Run-Matrix: OPTN, F6, F3 STAT, F3 DIST, F5 BINOMIAL: BinomialPD(k, n, p) and BinomialCD(lower, upper, n, p)
  • BinomialCD(k, n, p) with one bound means P(X ≤ k)

Where the Casio fx-CG50 trips people up

  • The order is k first, then n, then p. On the dialog it is labelled; in Run-Matrix it is not.
  • If the dialog shows Data: List it wants a list of x values. Switch to Variable with F2.
  • For the complement 1 − P(X ≤ 4) you can type 1 − Ans on the Run-Matrix screen right after the dialog result.

TI-84 Plus CE

In one line. 2nd VARS (DISTR): A binompdf(n, p, k) and B binomcdf(n, p, k). The cdf has ONE bound (P(X ≤ k)), so "between" and "at least" need a subtraction.

  1. 1

    Open the distribution menu and scroll down.

    2ndVARS (DISTR)A: binompdf(
  2. 2

    For exactly 4: trials 20, p 1/6, x value 4. Paste and ENTER.

    trials: 20p: 1/6x value: 4PasteENTER
  3. 3

    binompdf(20,1/6,4) = 0.2022417. Write 0.202.

  4. 4

    For at most 4 use binomcdf with x value 4.

    2ndVARS (DISTR)B: binomcdf(trials: 20p: 1/6x value: 4PasteENTER
  5. 5

    binomcdf(20,1/6,4) = 0.7687492. Write 0.769.

  6. 6

    For between 3 and 7 inclusive compute binomcdf(20,1/6,7) − binomcdf(20,1/6,2) = 0.66009. For at least 5 compute 1 − binomcdf(20,1/6,4) = 0.23125.

Typed form

  • binompdf(n, p, k)
  • binomcdf(n, p, k) = P(X ≤ k)
  • P(a ≤ X ≤ b) = binomcdf(n, p, b) − binomcdf(n, p, a − 1)

Where the TI-84 Plus CE trips people up

  • The subtraction for "between" uses a − 1, not a. P(3≤X≤7)=P(X≤7)−P(X≤2)P(3 \le X \le 7) = P(X \le 7) - P(X \le 2).
  • Leaving out the x value in binompdf returns the whole list of probabilities for k = 0 to n. Useful for a table, confusing in a hurry.
  • The wizard only appears if STAT WIZARDS is ON in MODE. If you see a bare binomcdf( on the home screen, type the three arguments separated by commas.

Check your own numbers

Type the values from your question. You get the keys with your numbers in them, the shaded picture to copy onto the paper, and the value to check against your screen.

01234567891011121314151617181920

X ~ B(20, 0.1667), P(X ≥ 5) = 1 − P(X ≤ 4) =

0.231 (0.23138)

Keys

menu6: Statistics5: DistributionsE: Binomial Cdfn: 20p: 0.1667Lower: 5Upper: 20OK

On the screen: binomCdf(20,0.1667,5,20)

Write: X ~ B(20, 0.1667), P(X ≥ 5) = 1 − P(X ≤ 4) = 0.231. Expected value E(X) = np = 3.33.

Where the marks go, on any calculator

  • "At least 5" is P(X≥5)=1−P(X≤4)P(X \ge 5) = 1 - P(X \le 4), not 1−P(X≤5)1 - P(X \le 5). The boundary value is the single most frequent lost mark in this topic.
  • "More than 5" is P(X≥6)P(X \ge 6); "fewer than 5" is P(X≤4)P(X \le 4). Translate the words into an inequality with ≤ before you type anything.
  • Not checking the binomial conditions. If the question says "without replacement" from a small group, it is not binomial, and the mark scheme will say so.
  • Using the pdf when the question says "at most". The pdf gives one value of kk only.

What to write on the paper

  1. State the distribution first: X∼B(20,16)X \sim B(20, \tfrac{1}{6}). This is usually a mark on its own.
  2. Write each probability in notation before the number: P(X≥5)=1−P(X≤4)=0.231P(X \ge 5) = 1 - P(X \le 4) = 0.231.
  3. For an "expected number" part, E(X)=npE(X) = np is written by hand, no calculator needed: here E(X)=20/6=3.33E(X) = 20/6 = 3.33.
  4. Give the unrounded calculator value (at least 4 s.f.) and then the 3 s.f. answer.

Questions students ask

Which calculator is easiest for binomial questions?
The TI-Nspire CX II and the Casio fx-CG50 take two bounds in the cumulative command, so "between" and "at least" need no subtraction. The TI-84 Plus CE takes one bound and you subtract. All three give the same answers; the TI-84 just needs one more line of working.
How do I find the smallest n so that P(X ≥ 1) > 0.99?
Rewrite as 1−(1−p)n>0.991 - (1-p)^n > 0.99 and solve for nn, either by hand with logarithms or by graphing y=1−(1−p)xy = 1 - (1-p)^x and finding where it crosses 0.99. The inverse binomial commands do not solve for nn; they solve for kk.
Does the IB want the calculator command in the working?
No. The IB wants the notation (P(X≤4)P(X \le 4)) and the value. Writing binomcdf(20,1/6,4) is tolerated but earns no mark that the notation does not earn.

Practise it

Other calculator skills

Written by Pietro Meloni, PhD, IB Mathematics and Physics tutor. Keystrokes checked on TI-Nspire CX II OS 6, Casio fx-CG50 OS 3 and TI-84 Plus CE OS 5; older systems may number the menus differently. Not affiliated with the International Baccalaureate Organization, Texas Instruments or Casio. Found a key that has moved? Write to pietro@pietromeloni.com.

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