Table of Contents

Chapter 17 - Probability Models

Wednesday, 4 January 2023
2-minute read
291 words

Geometric Probability Model

Peopel with O-negative blood are called "universal donors" because O-negative blood can be given to anyone else. Only about 6% of poeple have O-negative blood. How many people would you expect to examine before finding someone wiht O-negative.

\[ \frac{1}{0.06} = 16.67\% \]

Here's why this is geometric:

  1. There are two outcomes: O-negative blood or NOT O-negative blood
  2. The probability of success is the same for each person
  3. Trials either need to be independent OR contain less thna 10% of hte total population
  4. Looking for the first successful outcome

Syntax

geompdf(p, x) - Geometric distribution at a point

geomcdf(p, x) - Cumulative distribution in an interval

Casio Calculator

Arguments are in reverse order.

GeoPD(x, p)

GeoCD(x, p)

For an interval $[a, b]$:

GeoCD(b - a, p) * (1 - GeoCD(a, p))

Standard Deviation Formula

\[ \sigma = \sqrt{\frac{1 - p}{p^2}} \]

Example

Sergio is a bowler who rolls strikes 60% of the time.

  1. P(first stirke is within his first 3 rolls) = GeoCD(3, 0.6) = 0.936
  2. P(first stirke is NOT within his first 3 rolls) = 1 - GeoCD(3, 0.6) = 0.064
  3. P(first strike is on fifth roll) = GeoPD(5, 0.6) = 0.0154
  4. P(first strike is a roll between his 2nd and 6th rolls inclusive) = 0.396

Binomial

A binomial situation varies from a geometric situation in that we are not looking for the first success, but we are looking for the number of successes out of a fixed number of trials.

Here's how a binomial situation qualifies:

  1. Two outcomes
  2. Fixed probability
  3. Independent events or less than 10% of the total population
  4. Looking for a certain number of successes

Syntax

BinomialCD(x, n, p)

Standard Deviation Formula

\[ \sigma = \sqrt{np(1 - p)} \]

n = number of trials