The Binomial Distribution For the very common case of “Either-Or” experiments with only two possible outcomes.

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Presentation transcript:

The Binomial Distribution For the very common case of “Either-Or” experiments with only two possible outcomes

Recognize Binomial Situations

A special kind of probability distribution OutcomesProbabilities One of the events The other event TotalExactly 1

The Binomial Probability Formula

Practice with the Formula

Summary of the 7-11 experiment X successesP(X successes) 0 times (no sevens or elevens) 1 time 2 times 3 times 4 times 5 times (all sevens and elevens) Total (must equal !!)

Sometimes you add probabilities Probability of at least three wins in five trials – P(X≥3) = P(X=3) + P(X=4) + P(X=5) add them up! Probability of more than three wins – P(X>3) = P(X=4) + P(X=5) Probability of at most three wins – P(X≤3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) Probability of fewer than three wins – P(X<3) = P(X=0) + P(X=1) + P(X=2)

Use the Complement to save time

TI-84 Computations

binomcdf(n, p, x) = P(X=0) + P(X=1) + … P(X=x) successes in n trials binomcdf(n, p, x) does lots of little binompdf() for you for x = 0, x = 1, etc. up to the x you told it, and it adds up the results The “cdf” in “binomcdf” stands for “cumulative distribution function”

Try and verify binomcdf(n,p,x) X successesP(X successes) Using binompdf P(0 thru X) successes Using binomcdf 0 times (no sevens or elevens) 1 time 2 times 3 times 4 times 5 times (all sevens and elevens) Total (must equal !!)

binomcdf() and complements Sevens or elevens, n = 50 trials again P(no more than 10 successes) – binomcdf(50, 8/36, 10) P(fewer than 10 successes) – binomcdf(50, 8/36, 9) P(more than 10 successes) – use complement! – 1 minus binomcdf(50, 8/36, 10) P(at least 10 successes) – use complement! – 1 minus binomcdf(50, 8/36, 9)

Mean, Variance, and Standard Deviation

Standard Deviation What happens to the standard deviation in the seven-eleven experiment as the number of trials, n, increases?

Advanced TI-84 Exercise Y 1 =binompdf(20,8/36,X) seq(X,X,0,20) STO> L 1 seq(Y 1 (X),X,0,20) STO> L 2 STAT PLOT for these two lists, histogram WINDOW Xmin=0, Xmax=20, Ymin=-0.1,Ymax=0.6 ZOOM 9:ZoomStat