Webworks Probability and Statistics for the Liberal Arts I Asnwers

Learning Outcomes

• Depict a sample space and simple and chemical compound events in it using standard notation
• Calculate the probability of an effect using standard annotation
• Calculate the probability of two contained events using standard note
• Recognize when two events are mutually sectional
• Calculate a conditional probability using standard notation

Probability is the likelihood of a item outcome or result happening. Statisticians and actuaries utilize probability to make predictions about events.  An actuary that works for a auto insurance company would, for instance, exist interested in how probable a 17 yr old male would exist to go in a auto blow.  They would use data from past events to make predictions about future events using the characteristics of probabilities, then use this data to summate an insurance charge per unit.

In this section, we will explore the definition of an outcome, and larn how to calculate the probability of it's occurance.  We will besides practise using standard mathematical note to summate and describe different kinds of probabilities.

Bones Concepts

If you roll a die, pick a card from deck of playing cards, or randomly select a person and observe their hair color, we are executing an experiment or procedure. In probability, nosotros look at the likelihood of different outcomes.

We begin with some terminology.

Events and Outcomes

• The result of an experiment is chosen an event.
• An event is any detail event or group of outcomes.
• A simple consequence is an event that cannot be broken down further
• The sample space is the set of all possible simple events.

example

If we scroll a standard 6-sided die, draw the sample space and some simple events.

Basic Probability

Given that all outcomes are equally likely, we can compute the probability of an consequence E using this formula:

[latex]P(E)=\frac{\text{Number of outcomes corresponding to the consequence E}}{\text{Total number of as-likely outcomes}}[/latex]

examples

If we gyre a half-dozen-sided die, calculate

1. P(rolling a 1)
2. P(rolling a number bigger than 4)

This video describes this instance and the previous one in detail.

Permit's say you have a bag with twenty cherries, 14 sweet and 6 sour. If you pick a red at random, what is the probability that it volition be sweet?

Show Solution

In that location are 20 possible cherries that could be picked, and so the number of possible outcomes is 20. Of these xx possible outcomes, 14 are favorable (sweet), and so the probability that the cherry will exist sweet is [latex]\frac{fourteen}{twenty}=\frac{vii}{ten}[/latex].
There is one potential complexity to this example, nonetheless. Information technology must be assumed that the probability of picking any of the cherries is the aforementioned every bit the probability of picking any other. This wouldn't be truthful if (let u.s.a. imagine) the sweet cherries are smaller than the sour ones. (The sour cherries would come to hand more readily when y'all sampled from the purse.) Let us keep in mind, therefore, that when we assess probabilities in terms of the ratio of favorable to all potential cases, we rely heavily on the assumption of equal probability for all outcomes.

Try It

At some random moment, you await at your clock and annotation the minutes reading.

a. What is probability the minutes reading is 15?

b. What is the probability the minutes reading is fifteen or less?

Cards

A standard deck of 52 playing cards consists of four suits (hearts, spades, diamonds and clubs). Spades and clubs are black while hearts and diamonds are cherry-red. Each suit contains 13 cards, each of a different rank: an Ace (which in many games functions as both a depression card and a high card), cards numbered 2 through 10, a Jack, a Queen and a King.

case

Compute the probability of randomly cartoon one card from a deck and getting an Ace.

This video demonstrates both this instance and the previous cherry instance on the folio.

Certain and Impossible events

• An impossible event has a probability of 0.
• A certain upshot has a probability of ane.
• The probability of whatsoever event must exist [latex]0\le P(E)\le 1[/latex]

Try It

In the course of this section, if you lot compute a probability and get an reply that is negative or greater than 1, you have made a mistake and should check your work.

Types of Events

Complementary Events

Now allow us examine the probability that an event does non happen. As in the previous section, consider the situation of rolling a half-dozen-sided die and commencement compute the probability of rolling a six: the answer is P(6) =one/6. Now consider the probability that we exercise not roll a vi: there are 5 outcomes that are non a half-dozen, so the respond is P(non a six) = [latex]\frac{five}{vi}[/latex]. Detect that

[latex]P(\text{six})+P(\text{non a half dozen})=\frac{i}{vi}+\frac{5}{6}=\frac{6}{6}=ane[/latex]

This is not a coincidence.  Consider a generic state of affairs with northward possible outcomes and an upshot E that corresponds to chiliad of these outcomes. Then the remaining due north – m outcomes correspond to E not happening, thus

[latex]P(\text{non}Due east)=\frac{n-m}{n}=\frac{due north}{n}-\frac{m}{northward}=1-\frac{grand}{northward}=one-P(E)[/latex]

Complement of an Issue

The complement of an event is the upshot "E doesn't happen"

• The notation [latex]\bar{Due east}[/latex] is used for the complement of event E.
• We tin can compute the probability of the complement using [latex]P\left({\bar{E}}\right)=1-P(E)[/latex]
• Notice also that [latex]P(E)=i-P\left({\bar{E}}\right)[/latex]

example

If y'all pull a random card from a deck of playing cards, what is the probability it is not a heart?

This state of affairs is explained in the post-obit video.

Try It

Probability of two independent events

instance

Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a caput on the coin and a 6 on the die.

The prior case contained two contained events. Getting a sure outcome from rolling a die had no influence on the consequence from flipping the coin.

Contained Events

Events A and B are independent events if the probability of Result B occurring is the aforementioned whether or not Event A occurs.

case

Are these events independent?

1. A fair coin is tossed 2 times. The two events are (one) first toss is a head and (two) second toss is a head.
2. The two events (one) "Information technology will rain tomorrow in Houston" and (2) "It will rain tomorrow in Galveston" (a city nearly Houston).
3. Y'all draw a card from a deck, then draw a 2d card without replacing the first.

When ii events are independent, the probability of both occurring is the product of the probabilities of the private events.

P(A and B) for independent events

If events A and B are independent, then the probability of both A and B occurring is

[latex]P\left(A\text{ and }B\right)=P\left(A\right)\cdot{P}\left(B\right)[/latex]

where P(A and B) is the probability of events A and B both occurring, P(A) is the probability of upshot A occurring, and P(B) is the probability of event B occurring

If you look back at the coin and die example from earlier, yous can see how the number of outcomes of the first consequence multiplied by the number of outcomes in the 2nd result multiplied to equal the full number of possible outcomes in the combined result.

example

In your drawer you lot have 10 pairs of socks, half dozen of which are white, and 7 tee shirts, 3 of which are white. If you randomly accomplish in and pull out a pair of socks and a tee shirt, what is the probability both are white?

Examples of joint probabilities are discussed in this video.

Try It

The previous examples looked at the probability of both events occurring. Now we will look at the probability of either event occurring.

instance

Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the money or a 6 on the dice.

P(A or B)

The probability of either A or B occurring (or both) is

[latex]P(A\text{ or }B)=P(A)+P(B)–P(A\text{ and }B)[/latex]

example

Suppose we draw 1 carte du jour from a standard deck. What is the probability that we go a Queen or a King?

See more almost this instance and the previous i in the following video.

In the last example, the events were mutually sectional, so P(A or B) = P(A) + P(B).

Effort It

example

Suppose we draw i card from a standard deck. What is the probability that nosotros get a red card or a King?

Endeavour It

In your drawer you have x pairs of socks, 6 of which are white, and vii tee shirts, 3 of which are white. If you reach in and randomly grab a pair of socks and a tee shirt, what the probability at least one is white?

Example

The table below shows the number of survey subjects who have received and not received a speeding ticket in the last twelvemonth, and the color of their car. Notice the probability that a randomly chosen person:

1. Has a cerise motorcar and got a speeding ticket
2. Has a red car or got a speeding ticket.

	Speeding ticket	No speeding ticket	Full
Cerise car	15	135	150
Non red car	45	470	515
Total	lx	605	665

This table case is detailed in the following explanatory video.

Endeavour It

Conditional Probability

In the previous department we computed the probabilities of events that were contained of each other. We saw that getting a certain result from rolling a dice had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time.

In this department, we will consider events thataredependent on each other, chosen conditional probabilities.

Conditional Probability

The probability the issue B occurs, given that event A has happened, is represented as

P(B | A)

This is read as "the probability of B given A"

For example, if you draw a carte from a deck, then the sample space for the next carte du jour drawn has changed, considering you are now working with a deck of 51 cards. In the following example we will show yous how the computations for events like this are different from the computations we did in the concluding department.

case

What is the probability that two cards drawn at random from a deck of playing cards will both be aces?

Conditional Probability Formula

If Events A and B are not independent, then

P(A and B) = P(A) · P(B | A)

example

If you pull 2 cards out of a deck, what is the probability that both are spades?

Try Information technology

Example

The table below shows the number of survey subjects who have received and not received a speeding ticket in the final year, and the color of their car. Discover the probability that a randomly called person:

1. has a speeding ticket given they accept a cherry motorcar
2. has a red car given they take a speeding ticket

Speeding ticket	No speeding ticket	Total
Red car	xv	135	150
Non red automobile	45	470	515
Full	60	605	665

These kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the provisional probability of y'all having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood.

View more than about conditional probability in the following video.

Case

If you draw two cards from a deck, what is the probability that y'all volition get the Ace of Diamonds and a blackness bill of fare?

These 2 playing card scenarios are discussed further in the following video.

Endeavour It

Example

A home pregnancy test was given to women, then pregnancy was verified through blood tests.  The following tabular array shows the home pregnancy test results.

Find

1. P(non significant | positive examination event)
2. P(positive test result | not pregnant)

	Positive examination	Negative exam	Full
Pregnant	70	4	74
Not Pregnant	5	14	19
Total	75	xviii	93

See more about this example hither.

Endeavor It

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Source: https://courses.lumenlearning.com/wmopen-mathforliberalarts/chapter/computing-the-probability-of-an-event/

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