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The Binomial Distribution

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

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The Binomial Distribution

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  1. The Binomial Distribution

  2. 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.

  3. To use the binomial distribution the following conditions must apply: F the number of trials must be fixed I each trial must be independent of the other S The probability of success at each trial must be constant T there are only two outcomes, success or failure

  4. We use the following parametersfor the binomial distribution: n is 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) x is the total number of successes in the trial π + 1 - π = 1 p+q=1

  5. 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 πis 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

  6. The formula for calculating binomial probabilities is: P(X=x) = 0≤x≤n xεW 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 So P(X=2) = 0.24576 x=2 n=6 p=1/5

  7. 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

  8. 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) = 0.98304 F5 Dist F5 BINM F2 Bcd F2 VAR

  9. Summary so far The only 2 options on GC so change all questions into one of these forms 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 4 3

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