**3.3 The binomial formula**

**3.3.3 Calculator: binomial probabilities**

**TI-83/84: COMPUTING THE BINOMIAL COEFFICIENT** n
x
n
x
n
x

Use MATH, PRB, nCr to evaluate n choose r. Here r and x are different letters for the same quantity.

1. Type the value ofn. 2. SelectMATH.

3. Right arrow toPRB. 4. Choose3:nCr. 5. Type the value ofx. 6. HitENTER.

Example: 5 nCr 3means 5 choose 3.

**CASIO FX-9750GII: COMPUTING THE BINOMIAL COEFFICIENT** n
x
n
x
n
x

1. Navigate to theRUN-MATsection (hitMENU, then hit1). 2. Enter a value forn.

3. Go toCATALOG(hit buttonsSHIFTand then7).

4. TypeC(hit thelnbutton), then navigate down to the boldedCand hit EXE. 5. Enter the value ofx. Example of what it should look like: 7C3.

6. HitEXE.

**TI-84: COMPUTING THE BINOMIAL FORMULA,**P(X =x) = n_{x}

px(1−p)n−x
P(X=x) = n_{x}

px_{(1}_{−}_{p}_{)}n−x
P(X=x) = n_{x}

px(1−p)n−x

Use2ND VARS,binompdfto evaluate the probability ofexactly xoccurrences out ofnindepen- dent trials of an event with probabilityp.

1. Select2ND VARS(i.e. DISTR)

2. ChooseA:binompdf(use the down arrow to scroll down). 3. Lettrialsben.

4. Letpbe p

5. Letx valuebe x.

6. SelectPasteand hitENTER.

TI-83: Do step 1, choose 0:binompdf, then entern,p, andxseparated by commas: binompdf(n, p, x). Then hitENTER.

**TI-84: COMPUTING**P(X≤x) = n
0
p0_{(1}_{−}_{p}_{)}n−0_{+}_{...}_{+} n
x
px_{(1}_{−}_{p}_{)}n−x
P(X ≤x) = n_{0}
p0(1−p)n−0+...+ n_{x}
px(1−p)n−x
P(X≤x) = n
0
p0_{(1}_{−}_{p}_{)}n−0_{+}_{...}_{+} n
x
px_{(1}_{−}_{p}_{)}n−x

Use2ND VARS,binomcdfto evaluate the cumulative probability ofat most xoccurrences out of nindependent trials of an event with probabilityp.

1. Select2ND VARS(i.e. DISTR)

2. ChooseB:binomcdf(use the down arrow). 3. Lettrialsben.

4. Letpbe p

5. Letx valuebe x.

6. SelectPasteand hitENTER.

TI-83: Do steps 1-2, then enter the values for n, p, and x separated by commas as follows: binomcdf(n, p, x). Then hitENTER.

**CASIO FX-9750GII: BINOMIAL CALCULATIONS**

1. Navigate toSTAT(MENU, then hit 2). 2. SelectDIST(F5), and thenBINM(F5).

3. Choose whether to calculate the binomial distribution for a specific number of successes, P(X =k), or for a rangeP(X≤k) of values (0 successes, 1 success, ...,xsuccesses).

• For a specific number of successes, chooseBpd(F1). • To consider the range 0, 1, ...,xsuccesses, chooseBcd(F1). 4. If needed, setDatato Variable(Varoption, which isF2).

5. Enter the value forx(x),Numtrial(n), andp(probability of a success). 6. HitEXE.

**GUIDED PRACTICE 3.71**

Find the number of ways of arranging 3 blue marbles and 2 red marbles.55

**GUIDED PRACTICE 3.72**

There are 13 marbles in a bag. 4 are blue and 9 are red. Randomly draw 5 marbleswith replacement. Find the probability you get exactly 3 blue marbles.56

**GUIDED PRACTICE 3.73**

There are 13 marbles in a bag. 4 are blue and 9 are red. Randomly draw 5 marbleswith replacement. Find the probability you getat most 3 blue marbles (i.e. less than or equal to 3 blue marbles).57

55_{Here}_{n}_{= 5 and}_{x}_{= 3. Doing 5}

nCr3 gives the number of combinations as 10.

56_{Here,}_{n}_{= 5,}_{p}_{= 4/13, and}_{x}_{= 3, so set}_{trials}_{= 5,}_{p}_{= 4/13 and}_{x value}_{= 3. The probability is 0.1396.}
57_{Similarly, set}_{trials}_{= 5,}_{p}_{= 4/13 and}_{x value}_{= 3. The cumulative probability is 0.9662.}

**Section summary**

• n

x

, thebinomial coefficient, describes the number of combinations for arrangingxsuccesses
amongntrials. n_{x}

= _{x!(n}n!_{−}_{x)!}, wheren! = 1×2×3×...n, and 0!=0.

• Thebinomial formulacan be used to find the probability that something happens exactly x times in n trials. Suppose the probability of a single trial being a success is p. Then the probability of observing exactlyxsuccesses innindependent trials is given by

_{n}
x
px(1−p)n−x = n!
x!(n−x)!p
x_{(1}
−p)n−x

• To apply the binomial formula, the events must be independent from trial to trial. Addi- tionally,n, the number of trials must be fixed in advance, and p, the probability of the event occurring in a given trial, must be the same for each trial.

• To use the binomial formula, first confirm that the binomial conditions are met. Next, identify the number of trialsn, the number of times the event is to be a “success”x, and the probability that a single trial is a successp. Finally, plug these three numbers into the formula to get the probability of exactlyxsuccesses inntrials.

• Thepx_{(1}_{−}_{p}_{)}n−x_{part of the binomial formula is the probability of just one combination. Since}
there are n_{x}combinations, we add px_{(1}_{−}_{p}_{)}n−x _{up} n

x

times. We can think of the binomial formula as: [# of combinations]×P(a single combination).

• To find a probability involvingat least or at most, first determine if the scenario is binomial. If so, apply the binomial formula as many times as needed and add up the results. e.g. P(at least 3 Heads in 5 tosses of a fair coin) = P(exactly 3 Heads) +P(exactly 4 Heads) + P(exactly 5 Heads), where each probability can be found using the binomial formula.

**Exercises**

**3.29 Exploring combinations.** A coin is tossed 5 times. How many sequences / combinations of Heads/Tails
are there that have:

(a) Exactly 1 Tail? (b) Exactly 4 Tails? (c) Eactly 3 Tails? (d) At least 3 Tails?

**3.30** **Political affiliation.** Suppose that in a large population, 51% identify as Democrat. A researcher
takes a random sample of 3 people.

(a) Use the binomial model to calculate the probability that two of them identify as Democrat.

(b) Write out all possible orderings of 3 people, 2 of whom identify as Democrat. Use these scenarios to calculate the same probability from part (a) but using the Addition Rule for disjoint events. Confirm that your answers from parts (a) and (b) match.

(c) If we wanted to calculate the probability that a random sample of 8 people will have 3 that identify as Democrat, briefly describe why the approach from part (b) would be more tedious than the approach from part (a).

**3.31** **Underage drinking, Part I.** Data collected by the Substance Abuse and Mental Health Services
Administration (SAMSHA) suggests that 69.7% of 18-20 year olds consumed alcoholic beverages in any
given year.58

(a) Suppose a random sample of ten 18-20 year olds is taken. Is the use of the binomial distribution appropriate for calculating the probability that exactly six consumed alcoholic beverages? Explain. (b) Calculate the probability that exactly 6 out of 10 randomly sampled 18- 20 year olds consumed an

alcoholic drink.

(c) What is the probability that exactly four out of ten 18-20 year olds have not consumed an alcoholic beverage?

(d) What is the probability that at most 2 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages?

(e) What is the probability that at least 1 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages?

**3.32 Chicken pox, Part I.**The National Vaccine Information Center estimates that 90% of Americans have
had chickenpox by the time they reach adulthood.59

(a) Suppose we take a random sample of 100 American adults. Is the use of the binomial distribution appropriate for calculating the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood? Explain.

(b) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood.

(c) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood?

(d) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox? (e) What is the probability that at most 3 out of 10 randomly sampled American adults have not had

chickenpox?

58_{SAMHSA, Office of Applied Studies,}_{National Survey on Drug Use and Health, 2007 and 2008}_{.}
59_{National Vaccine Information Center,}_{Chickenpox, The Disease & The Vaccine Fact Sheet}_{.}