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