R

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