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The Binomial Distribution. For the very common case of “Either-Or” experiments with only two possible outcomes. Recognize Binomial Situations. Only two possible outcomes in each trial. Probability for one of the outcomes. Probability for the other outcome.

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the binomial distribution

The Binomial Distribution

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

recognize binomial situations
Recognize Binomial Situations
  • Only two possible outcomes in each trial.
    • Probability for one of the outcomes.
    • Probability for the other outcome.
  • Some definite number of trials, .
    • They’re independent trials. don’t change.
  • We’re interested in , probability of a certain count of how many times event happens in those trials.
a special kind of probability distribution
A special kind of probability distribution
  • It’s the familiar probability distribution
  • But only two rows for the two outcomes.
  • Note that
    • Because probabilities must always sum to 1.00000
    • And this leads to .
the binomial probability formula
The Binomial Probability Formula
  • Question: If we have trials, what is the probability of occurrences of the “success” event (the one with probability )
  • Answer:
practice with the formula
Practice with the Formula
  • Experiment: Roll two dice
  • Event of interest: “I rolled a 7 or an 11”
  • Probability of success: (from
  • Probability of failure:
  • Number of trials
practice with the formula1
Practice with the Formula
  • Find P(2) successes in the seven/eleven game
  • How about 3 successes?
sometimes you add probabilities
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
Use the Complement to save time
  • Example:
  • “Probability of at least 3 wins”
  • P(X≥3) = P(X=3) + P(X=4) + … + P(X=49) + P(X=50)
  • This means 48 calculations and sum results.
  • EASIER: The complement is “fewer than 3”
  • Take 1 – [ P(X=0) + P(X=1) + P(X=2) ]
ti 84 computations
TI-84 Computations
  • binompdf(n, p, X) = probability of X successes in n trials
  • Recompute the table and make sure we get the same results as the by-hand calculations.
  • The “pdf” in “binompdf” stands for “probability distribution function”
ti 84 computations1
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”
binomcdf and complements
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
Mean, Variance, and Standard Deviation
  • We had formulas and methods for probability distributions in general.
  • The special case of the Binomial Probability Distribution has special shortcut formulas
    • Mean =
    • Variance =
    • Standard deviation =
mean variance and standard deviation1
Mean, Variance, and Standard Deviation
  • Compute these for the seven-eleven experiment with n = 5 trials
    • Mean =
    • Variance =
    • Standard deviation =
mean variance and standard deviation2
Mean, Variance, and Standard Deviation
  • Compute these for the seven-eleven experiment with n = 50 trials
    • Mean =
    • Variance =
    • Standard deviation =
mean variance and standard deviation3
Mean, Variance, and Standard Deviation
  • Compute these for the seven-eleven experiment with n = 100 trials
    • Mean = and Standard deviation =
  • “Expected value” – in 100 tosses of two dice, how many seven-elevens are expected?
    • Remember, the mean of a probability distribution is also called the “expected value”
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
Advanced TI-84 Exercise
  • Y1=binompdf(20,8/36,X)
  • seq(X,X,0,20) STO> L1
  • seq(Y1(X),X,0,20) STO> L2
  • STAT PLOT for these two lists, histogram
  • WINDOW Xmin=0, Xmax=20, Ymin=-0.1,Ymax=0.6
  • ZOOM 9:ZoomStat