The Binomial Distribution

Situations often arises where there are only two outcomes (which we label as success or failure). When this occurs we get a binomial distribution.

ExampleIn a multi-choice test, Sally guesses

the answers to the last 6 questions. Each question has 5 choices. The binomial distribution describes the probability of 0, 1, 2, etc successes out of the 6 number of trials.

To use the binomial distribution the following conditions must apply:

F the number of trials must be fixedI each trial must be independent of

the otherS The probability of success at each

trial mustbe constant

T there are only two outcomes, success or

failure

We use the following parametersfor the binomial distribution:

n is the number of trials conductedπ is the probability of success (can also use p)1 - π is the probability of failure (can also use q)x is the total number of successes

in thetrial

π + 1 - π = 1

p+q=1

Example: Give the values of n, π, 1- π and x for:

In a multi-choice test, Sally guesses the answers to the last 6 questions. Each question has 5 choices.

What is the probability that Sally guesses two out of the six correctly?

n is the number of trials conductedπ is the probability of success1 - π is the probability of failure alsox is the total number of successes

in the trial

n=6

1/5

4/5

x=2

The formula for calculating binomial probabilities is:

P(X=x) = 0≤x≤n xεW

But we can use our GC:2=StatsF5=dist F5 =Binm F1 = Bpd (since we are using = a single,

precise number)F2=Var to get screen with:

x numtrial p

)()1( xnx

x

n

x=2n=6

p=1/5

So P(X=2) = 0.24576

On the GC we use F2= Bcd for cumulative values ie when calculating ≤ (instead of = )

Example Two What is the probability that Sally gets 2 or less questions correct

P(X ≤2) = 2=Stats F5=dist F5 =Binm F2 = Bcd (since we are using ≤ more than one number – cumulative situation)F2=Var to get screen with:

x numtrial p

P(X ≤2) =0.90111

On the GC we must always turn < into ≤ questions

Example Two Find the probability that Sally gets less

than 4 questions correct

Find P(X<4) for n=6 and p=0.2 becomes:

P(X≤3) for n=6 and p=0.2P(X≤3) = 0.98304

F5 DistF5 BINMF2 BcdF2 VAR

Summary so far

Use your GC for Binomial distribution by using:

• Bpd for P(X= )• Bcd for P(X≤ )

• If P(X < ) change into P(X ≤ ) and use Bcd

The only 2 options on GC so

change all questions into one of these

forms

4 3