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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

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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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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