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Key equations
probability of an event with equally likely outcomes
probability of the union of two events
probability of the union of mutually exclusive events
probability of the complement of an event
Key concepts
Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
The probabilities in a probability model must sum to 1. See
[link] .
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 the sample space for the experiment. See
[link] .
To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See
[link] .
To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See
[link] .
The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See
[link] .
In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See
[link] .
Section exercises
Verbal
What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.
probability; The probability of an event is restricted to values between
and
inclusive of
and
The
union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the
union of two events from a probability model? How is it different?
The probability of the
union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets
and a union of events
the union includes either
or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between
and
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
