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In this section, you will:
  • Construct probability models.
  • Compute probabilities of equally likely outcomes.
  • Compute probabilities of the union of two events.
  • Use the complement rule to find probabilities.
  • Compute probability using counting theory.
Spaghetti map of the possible paths for a hurricane over the Southeastern United States
An example of a “spaghetti model,” which can be used to predict possible paths of a tropical storm. The figure is for illustrative purposes only and does not model any particular storm.

Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in [link] . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

Constructing probability models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment    , or an activity with an observable result. The numbers on the cube are possible results, or outcomes    , of this experiment. The set of all possible outcomes of an experiment is called the sample space    of the experiment. The sample space for this experiment is { 1 , 2 , 3 , 4 , 5 , 6 } . An event    is any subset of a sample space.

The likelihood of an event is known as probability    . The probability of an event p is a number that always satisfies 0 p 1 , where 0 indicates an impossible event and 1 indicates a certain event. A probability model    is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like [link] .

Outcome Probability
Winning the raffle 1%
Losing the raffle 99%

The sum of the probabilities listed in a probability model must equal 1, or 100%.

Given a probability event where each event is equally likely, construct a probability model.

  1. Identify every outcome.
  2. Determine the total number of possible outcomes.
  3. Compare each outcome to the total number of possible outcomes.

Constructing a probability model

Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.

Begin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: 1, 2, 3, 4, 5, and 6. There are six possible outcomes that make up the sample space.

Assign probabilities to each outcome in the sample space by determining a ratio of the outcome to the number of possible outcomes. There is one of each of the six numbers on the cube, and there is no reason to think that any particular face is more likely to show up than any other one, so the probability of rolling any number is 1 6 .

Outcome Roll of 1 Roll of 2 Roll of 3 Roll of 4 Roll of 5 Roll of 6
Probability 1 6 1 6 1 6 1 6 1 6 1 6
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Do probabilities always have to be expressed as fractions?

No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1.

Questions & Answers

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Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
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simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
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what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
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litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
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Practice Key Terms 9

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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