13.7 Probability (Page 3/18)

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A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

$\frac{7}{13}$

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Computing the probability of mutually exclusive events

Suppose the spinner in [link] is spun again, but this time we are interested in the probability of spinning an orange or a $d .$ There are no sectors that are both orange and contain a $d,$ so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is

P (E \cup F) = P (E) + P (F)

Notice that with mutually exclusive events, the intersection of $E$ and $F$ is the empty set. The probability of spinning an orange is $\frac{3}{6} = \frac{1}{2}$ and the probability of spinning a $d$ is $\frac{1}{6} .$ We can find the probability of spinning an orange or a $d$ simply by adding the two probabilities.

\begin{array}{l} P (E \cup^{​} F) = P (E) + P (F) \\ = \frac{1}{2} + \frac{1}{6} \\ = \frac{2}{3} \end{array}

The probability of spinning an orange or a $d$ is $\frac{2}{3} .$

A General Note

Probability of the union of mutually exclusive events

The probability of the union of two mutually exclusive events $E and F$ is given by

P (E \cup F) = P (E) + P (F)

How To

Given a set of events, compute the probability of the union of mutually exclusive events.

Determine the total number of outcomes for the first event.
Find the probability of the first event.
Determine the total number of outcomes for the second event.
Find the probability of the second event.
Add the probabilities.

Computing the probability of the union of mutually exclusive events

A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.

The events “drawing a heart” and “drawing a spade” are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is $\frac{1}{4},$ and the probability of drawing a spade is also $\frac{1}{4},$ so the probability of drawing a heart or a spade is

\frac{1}{4} + \frac{1}{4} = \frac{1}{2}

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A card is drawn from a standard deck. Find the probability of drawing an ace or a king.

$\frac{2}{13}$

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Using the complement rule to compute probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event $E,$ denoted $E^{'},$ is the set of outcomes in the sample space that are not in $E .$ For example, suppose we are interested in the probability that a horse will lose a race. If event $W$ is the horse winning the race, then the complement of event $W$ is the horse losing the race.

To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.

P (E^{'}) = 1 - P (E)

The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is $\frac{1}{9},$ the probability of the horse losing the race is simply

1 - \frac{1}{9} = \frac{8}{9}

A General Note

The complement rule

The probability that the complement of an event will occur is given by

P (E^{'}) = 1 - P (E)

Using the complement rule to calculate probabilities

Two six-sided number cubes are rolled.

Find the probability that the sum of the numbers rolled is less than or equal to 3.
Find the probability that the sum of the numbers rolled is greater than 3.

The first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are $6 \times 6,$ or $36$ total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.

$1-1$	$1-2$	$1-3$	$1-4$	$1-5$	$1-6$
$2-1$	$2-2$	$2-3$	$2-4$	$2-5$	$2-6$
$3-1$	$3-2$	$3-3$	$3-4$	$3-5$	$3-6$
$4-1$	$4-2$	$4-3$	$4-4$	$4-5$	$4-6$
$5-1$	$5-2$	$5-3$	$5-4$	$5-5$	$5-6$
$6-1$	$6-2$	$6-3$	$6-4$	$6-5$	$6-6$

We need to count the number of ways to roll a sum of 3 or less. These would include the following outcomes: 1-1, 1-2, and 2-1. So there are only three ways to roll a sum of 3 or less. The probability is
$\frac{3}{36} = \frac{1}{12}$
Rather than listing all the possibilities, we can use the Complement Rule. Because we have already found the probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that the sum of the numbers rolled is greater than 3.
$\begin{array}{l} P (E^{'}) = 1 - P (E) \\ = 1 - \frac{1}{12} \\ = \frac{11}{12} \end{array}$

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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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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