Grabbing From the Bag Again Probability

Learning Outcomes

• Describe a sample space and uncomplicated and compound events in it using standard notation
• Summate the probability of an event using standard notation
• Calculate the probability of 2 contained events using standard notation
• Recognize when 2 events are mutually exclusive
• Calculate a provisional probability using standard notation

Probability is the likelihood of a item consequence or outcome happening. Statisticians and actuaries utilise probability to make predictions about events.  An actuary that works for a car insurance company would, for example, be interested in how likely a 17 year old male would be to become in a car accident.  They would employ data from past events to make predictions virtually future events using the characteristics of probabilities, so apply this information to calculate an insurance rate.

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

Basic Concepts

If you lot 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, we await at the likelihood of different outcomes.

We begin with some terminology.

Events and Outcomes

• The result of an experiment is called an upshot.
• An event is whatsoever detail issue or group of outcomes.
• A simple upshot is an event that cannot be cleaved down further
• The sample space is the set of all possible simple events.

example

If nosotros roll a standard half-dozen-sided die, describe the sample infinite and some uncomplicated events.

Bones Probability

Given that all outcomes are equally likely, nosotros can compute the probability of an event East using this formula:

[latex]P(Due east)=\frac{\text{Number of outcomes corresponding to the event East}}{\text{Total number of equally-likely outcomes}}[/latex]

examples

If we ringlet a 6-sided die, calculate

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

This video describes this example and the previous one in item.

Permit's say yous have a bag with xx cherries, xiv sweetness and 6 sour. If you lot choice a ruby at random, what is the probability that it will exist sweet?

Evidence Solution

There are 20 possible cherries that could exist picked, and so the number of possible outcomes is 20. Of these 20 possible outcomes, 14 are favorable (sweet), so the probability that the red volition be sweet is [latex]\frac{14}{20}=\frac{7}{10}[/latex].
There is one potential complication to this example, withal. It must exist assumed that the probability of picking whatever of the cherries is the same as the probability of picking any other. This wouldn't exist true if (permit us imagine) the sugariness cherries are smaller than the sour ones. (The sour cherries would come to hand more readily when you sampled from the bag.) Permit united states of america keep in listen, therefore, that when we appraise probabilities in terms of the ratio of favorable to all potential cases, nosotros rely heavily on the assumption of equal probability for all outcomes.

Effort It

At some random moment, y'all look at your clock and note the minutes reading.

a. What is probability the minutes reading is 15?

b. What is the probability the minutes reading is 15 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 ruby. Each suit contains thirteen cards, each of a different rank: an Ace (which in many games functions as both a low menu and a high card), cards numbered two through 10, a Jack, a Queen and a King.

example

Compute the probability of randomly drawing 1 carte from a deck and getting an Ace.

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

Certain and Impossible events

• An impossible outcome has a probability of 0.
• A sure upshot has a probability of 1.
• The probability of any event must be [latex]0\le P(E)\le 1[/latex]

Try It

In the form of this section, if y'all compute a probability and get an answer that is negative or greater than one, you have fabricated a fault and should check your work.

Types of Events

Complementary Events

Now let us examine the probability that an issue does not happen. Equally in the previous section, consider the situation of rolling a half-dozen-sided dice and start compute the probability of rolling a half-dozen: the reply is P(6) =1/6. Now consider the probability that we practice not roll a half-dozen: there are 5 outcomes that are not a half dozen, so the answer is P(non a six) = [latex]\frac{5}{6}[/latex]. Notice that

[latex]P(\text{half-dozen})+P(\text{non a six})=\frac{1}{six}+\frac{v}{vi}=\frac{six}{vi}=1[/latex]

This is not a coincidence.  Consider a generic situation with n possible outcomes and an event E that corresponds to m of these outcomes. And so the remaining n – m outcomes represent to E non happening, thus

[latex]P(\text{not}East)=\frac{n-m}{n}=\frac{n}{due north}-\frac{m}{n}=1-\frac{m}{n}=one-P(East)[/latex]

Complement of an Consequence

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

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

case

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

This situation is explained in the following video.

Try It

Probability of ii independent events

example

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

The prior example contained two contained events. Getting a certain effect from rolling a die had no influence on the outcome from flipping the coin.

Independent Events

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

example

Are these events independent?

1. A fair coin is tossed two times. The two events are (1) commencement toss is a caput and (2) second toss is a head.
2. The two events (one) "Information technology volition rain tomorrow in Houston" and (2) "Information technology volition rain tomorrow in Galveston" (a city near Houston).
3. You draw a card from a deck, and then draw a second card without replacing the starting time.

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 contained 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 event A occurring, and P(B) is the probability of event B occurring

If you lot look dorsum at the coin and die example from earlier, you tin can encounter how the number of outcomes of the first event multiplied by the number of outcomes in the 2d event multiplied to equal the full number of possible outcomes in the combined issue.

example

In your drawer y'all have 10 pairs of socks, 6 of which are white, and 7 tee shirts, three of which are white. If you randomly reach 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.

Attempt It

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

example

Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the coin or a vi 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 describe one card from a standard deck. What is the probability that we get a Queen or a Male monarch?

See more about this example and the previous i in the following video.

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

Try It

example

Suppose we depict one bill of fare from a standard deck. What is the probability that nosotros become a cherry-red menu or a King?

Try Information technology

In your drawer you take 10 pairs of socks, 6 of which are white, and 7 tee shirts, iii of which are white. If you reach in and randomly take hold of a pair of socks and a tee shirt, what the probability at least 1 is white?

Example

The tabular array beneath 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. Detect the probability that a randomly chosen person:

1. Has a red car and got a speeding ticket
2. Has a cherry-red machine or got a speeding ticket.

	Speeding ticket	No speeding ticket	Total
Red car	15	135	150
Non ruby-red machine	45	470	515
Total	lx	605	665

This table instance is detailed in the following explanatory video.

Endeavor Information technology

Conditional Probability

In the previous department we computed the probabilities of events that were independent of each other. We saw that getting a sure outcome from rolling a die had no influence on the outcome from flipping a money, fifty-fifty though we were computing a probability based on doing them at the same fourth dimension.

In this section, we will consider events thataredependent on each other, called conditional probabilities.

Conditional Probability

The probability the effect B occurs, given that event A has happened, is represented every bit

P(B | A)

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

For case, if you draw a card from a deck, then the sample space for the next card drawn has changed, considering you are now working with a deck of 51 cards. In the post-obit example we will bear witness you how the computations for events like this are different from the computations we did in the last section.

example

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

Conditional Probability Formula

If Events A and B are non 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?

Endeavor It

Instance

The tabular array below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the colour of their car. Detect the probability that a randomly chosen person:

1. has a speeding ticket given they take a blood-red car
2. has a cherry car given they have a speeding ticket

Speeding ticket	No speeding ticket	Total
Red car	15	135	150
Not scarlet car	45	470	515
Total	60	605	665

These kinds of conditional probabilities are what insurance companies use to make up one's mind your insurance rates. They look at the conditional probability of you having blow, given your historic period, your car, your automobile colour, your driving history, etc., and price your policy based on that likelihood.

View more than about conditional probability in the following video.

Instance

If you describe two cards from a deck, what is the probability that yous volition get the Ace of Diamonds and a blackness menu?

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

Endeavour It

Example

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

Find

1. P(non pregnant | positive test result)
2. P(positive exam outcome | not pregnant)

	Positive test	Negative test	Total
Pregnant	lxx	4	74
Not Significant	5	14	19
Total	75	18	93

See more almost this example here.

Attempt It

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

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