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. Example In 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: Fthe number of trials must be fixed Ieach trial must be independent of the other SThe probability of success at each trial must be constant Tthere are only two outcomes, success or failure

We use the following parametersfor the binomial distribution: nis the number of trials conducted π is the probability of success (can also use p) 1 - π is the probability of failure (can also use q) xis the total number of successes in the trial π 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 conducted π is the probability of success 1 - π is the probability of failure also x 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 x ε W But we can use our GC: 2=Stats F5=dist F5 =Binm F1 = Bpd (since we are using = a single, precise number) F2=Var to get screen with: x numtrial p x=2 n=6 p=1/5 So P(X=2) =

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) =

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=0.2 P(X≤3) = F5 Dist F5 BINM F2 Bcd F2 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 43