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Key equations
probability of an event with equally likely outcomes
$$P(E)=\frac{n(E)}{n(S)}$$
probability of the union of two events
$$P(E\cup F)=P(E)+P(F)-P(E\cap F)$$
probability of the union of mutually exclusive events
$$P(E\cup F)=P(E)+P(F)$$
probability of the complement of an event
$$P(E\text{'})=1-P(E)$$
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
$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}$ inclusive of
$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$
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
$\text{\hspace{0.17em}}A\text{}\text{and}B\text{\hspace{0.17em}}$ and a union of events
$\text{\hspace{0.17em}}A\text{and}B,\text{\hspace{0.17em}}$ the union includes either
$\text{\hspace{0.17em}}A\text{or}B\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0
then
4x = 2-3
4x = -1
x = -(1÷4) is the answer.
Jacob
4x-2+3
4x=-3+2
4×=-1
4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3
4x=-3+2
4x=-1
4x÷4=-1÷4
x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?