6.2: Binomial and Geometric Probability

Binomial Distributions

  • A setting is considered binomial if the four following criteria are met   * B: Binary     * Trials must be able to be categorized as successes or failures   * I: Independence     * Trials must be independent from one another   * N: Number     * There must be a fixed number of trials   * S: Success     * There must be a constant probability of success for each trial (represented by variable p)
  • Binomial random variables meet all four of these conditions, and are described by binomial distributions
  • B(n,p) means that there is a binomial distribution where x counts the number of successes   * n = number of trials/observations   * p = probability of success
  • x can only take on whole number values from 0 to n
  • Mean and standard deviation for binomial random variables   * µ = np   * σ = square root of np(1-p)
Formula for Binomial Predictability
  • This means that:   * Out of n trials, there are k successes   * n choose k counts the number of ways to have successes in n trials   * p^k calculates the number of times k succeeds   * (1-p)^(n-k) calculates the number of failures
  • Calculator use   * Particular success = binomPdf     * Eg. P (x=5)   * Cumulative success = binomCdf     * Eg. P (x<5)     * P(1<x<12)   * Menu 6→5→D/E   * Must show equations + work on paper to reflect what is done on calculator for credit on tests + AP exam
Example
  • The probability of hitting a bullseye while playing darts is 0.37
  • We are going to throw 25 darts and count the number of bullseyes.
  • Each throw is independent.

 

Geometric Distributions

  • Meet all of the criteria for binomial probabilities except there is no set number of trials → they are continued until failure
  • If x is geometric, p is the probability of success, and 1-p is the probability of failure, then:   * P(first success on nth trial) = (1-p)^(n-1) x (p)^1     * We succeeded once (on the last trial) and failed every time before that   * P(success takes more than n trials) = (1-p)^n     * We only know we failed this many times
Example

 

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