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11.7 Probability  (Page 2/18)

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Construct a probability model for tossing a fair coin.

Outcome Probability
Roll of 1
Roll of 2
Roll of 3
Roll of 4
Roll of 5
Roll of 6
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Computing probabilities of equally likely outcomes

Let   S   be a sample space for an experiment. When investigating probability, an event is any subset of   S .   When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in   S .   Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in   S ,   so the probability of the event is   4 6 = 2 3 .  

Computing the probability of an event with equally likely outcomes

The probability of an event E in an experiment with sample space S with equally likely outcomes is given by

  P ( E ) = number of elements in  E number of elements in  S = n ( E ) n ( S )  

  E is a subset of S , so it is always true that 0 ≤ P ( E ) ≤ 1.  

Computing the probability of an event with equally likely outcomes

A number cube is rolled. Find the probability of rolling an odd number.

The event “rolling an odd number” contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of the event.

  P ( E ) = 3 6 = 1 2  
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A number cube is rolled. Find the probability of rolling a number greater than 2.

  2 3  

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Computing the probability of the union of two events

We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events       E  and  F , written  E ∪ F ,   is the event that occurs if either or both events occur.

  P ( E ∪ F ) = P ( E ) + P ( F ) − P ( E ∩ F )  

Suppose the spinner in [link] is spun. We want to find the probability of spinning orange or spinning a   b .  

A pie chart with six pieces with two a's colored orange, one b colored orange and another b colored red, one d colored blue, and one c colored green.

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is   3 6 = 1 2 .   There are a total of 6 sections, and 2 of them have a   b .   So the probability of spinning a   b is 2 6 = 1 3 . If we added these two probabilities, we would be counting the sector that is both orange and a b twice. To find the probability of spinning an orange or a b , we need to subtract the probability that the sector is both orange and has a b .

  1 2 + 1 3 − 1 6 = 2 3  

The probability of spinning orange or a b   is 2 3 .

Probability of the union of two events

The probability of the union of two events E and F (written   E ∪ F ) equals the sum of the probability of E and the probability of F minus the probability of E and F occurring together ( which is called the intersection of E and F and is written as E ∩ F ).

  P ( E ∪ F ) = P ( E ) + P ( F ) − P ( E ∩ F )  

Computing the probability of the union of two events

A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.

A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is   1 4 .   There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing a 7 is   1 13 .

The only card in the deck that is both a heart and a 7 is the 7 of hearts, so the probability of drawing both a heart and a 7 is   1 52 .   Substitute   P ( H ) = 1 4 ,   P ( 7 ) = 1 13 ,   and   P ( H ∩ 7 ) = 1 52   into the formula.

P ( E ∪ ​ F ) = P ( E ) + P ( F ) − P ( E ∩ ​ F )                 = 1 4 + 1 13 − 1 52                 = 4 13

The probability of drawing a heart or a 7 is   4 13 .

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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, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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