Which of the Following Describes a Simple Event

Which of the Following Describes a Simple Event.

In this explainer, we will learn how to find the probability of a simple effect.

The probability of an result is the likelihood of information technology occurring.

When we discuss the likelihood of an event happening in everyday life, we may use some common words to describe this likelihood, for example, “certain”, “likely”, “very unlikely”, or “impossible”. In mathematics, we can assign a numerical value to a probability. Impossible events have a probability of 0, and events that are sure to happen accept a probability of 1. Events that are every bit likely can be written with a probability of 0.5, or

ane 2 .

The sum of the probabilities of all possible outcomes must equal 1. For instance, when flipping a coin, the probability of getting “heads” plus the probability of getting “tails” is i. This is because the probability of getting either one of heads or tails is certain, that is, a probability of ane.

In probability terms, a simple event refers to an event with a unmarried outcome, for example, getting “heads” with a single toss of a money, or rolling a four on a die.

We also need to consider “fairness” when discussing probability.

Consider the state of affairs of flipping a fair coin. Information technology can be described every bit fair as the outcomes are every bit likely. If the coin has sides “heads” and “tails”, so the outcome of getting “tails” would be 1 outcome out of a possible 2 outcomes. We could write this as a fraction,

1 2 .

Definition: Probability of a Simple Issue

The probability of a simple event is
p r o b a b i l i t y o f a n e v e n t n u chiliad b e r o f o u t c o m e s w h e r e t h e e v e n t o c c u r s t o t a l north u m b e r o f p o s s i b l east o u t c o m e s = .

Commonly in probability, nosotros may use the notation
𝑃 ( ) e v e northward t
to represent the probability of an outcome occurring. For example, when selecting a green or blue brawl from a bag,
𝑃 ( ) g r e e n
can be used to represent the probability of selecting a green ball.

We can at present meet how this data can exist applied in a number of dissimilar examples.

Example 1: Determining the Theoretical Probability of an Event

A class has xviii boys and ix girls. What is the probability that a randomly selected educatee is a daughter?

Reply

We tin can recall that the probability of a elementary event can be written as
p r o b a b i l i t y o f a n e v e n t due north u m b e r o f o u t c o m e s w h e r e t h e e v e n t o c c u r south t o t a 50 n u m b e r o f p o s s i b fifty east o u t c o g e s = .
In this case, we need to calculate the probability of selecting a girl, which we tin can write as

𝑃 ( ) k i r l .

We can write the statement
𝑃 ( ) = . thousand i r l n u m b e r o f yard i r l s t o t a 50 n u m b due east r o f due south t u d e n t s

As there are 18 boys and ix girls in the class, and so the total number of students must be

i 8 + 9 = 2 7 .
Substituting the information that the number of girls = 9 and the total number of students = 27 gives us
𝑃 ( ) = ix two seven . thou i r l

Simplifying this fraction, we take
𝑃 ( ) = 1 3 . g i r fifty

Thus, the probability that a randomly selected pupil is a girl is

1 3 .

In the post-obit instance, we volition see how we may oftentimes demand to use physical information about an object to obtain the likelihood of an event happening, for example, past examining the sections of a spinner.

Example 2: Determining the Probability of an Event Involving a Spinner

What is the probability of the pointer landing on an even number when the given spinner is spun?

Answer

We consider that in this spinner, every bit the sections are of equal size, then there is an equal probability of the spinner landing in each department, assuming that the spinner is fair.

Nosotros recollect that the probability of a uncomplicated event is given by
p r o b a b i l i t y o f a due north e v east n t northward u m b eastward r o f o u t c o yard east south w h e r e t h eastward e five due east n t o c c u r south t o t a l n u m b due east r o f p o south s i b l eastward o u t c o m east s = .

To find the probability of landing on an even number,

𝑃 ( ) east v e n ,
nosotros tin write
𝑃 ( ) = . e v e n n u m b e r o f e 5 east n 5 a l u e s o north t h east s p i n n east r t o t a l northward u m b e r o f s e c t i o due north s o n t h east south p i n n e r

We consider the even and odd values on the spinner. Even numbers are integers that are divisible by 2. Odd numbers are integers that are not divisible past 2. Equally 12 and 14 are the only even numbers on the spinner, so the number of possible outcomes that are even is 2. The total number of outcomes is the full number of sections on the spinner, eight. We tin can substitute these values into our equation, giving
𝑃 ( ) = 2 8 . e v east n

Simplifying the fraction, we accept
𝑃 ( ) = i four . e v due east n

Therefore, the probability of the arrow landing on an even number when the spinner is spun is

i four .

Nosotros consider another example.

Example 3: Determining the Probability of an Upshot

A deck of cards contains cards numbered from 1 to 81. If a card is picked at random, what is the probability of picking a card with a number that is divisible past five?

Answer

We can consider the deck of cards as follows.

In order to notice the probability of picking a particular card or blazon of card, we call up that the probability of a unproblematic consequence can be given as
p r o b a b i l i t y o f a due north east 5 e n t n u 1000 b e r o f o u t c o k e s w h e r e t h e east v e due north t o c c u r due south t o t a l n u chiliad b due east r o f p o s s i b l due east o u t c o 1000 east southward = .

For the issue of picking a carte du jour that is divisible by v,

𝑃 ( 5 ) d i v i s i b l e b y ,
we could write the equation that
𝑃 ( v ) = five . d i v i s i b l e b y n u m b due east r o f c a r d v a l u e south d i 5 i s i b l east b y t o t a 50 due north u grand b e r o f c a r d s

We recall that divisible means to be able to divide by a number and get an integer answer. Numbers that are divisible by five are too multiples of 5. We can listing the numbers that are divisible by 5, between 1 and 81, as
5 , i 0 , 1 five , ii 0 , 2 5 , 3 0 , 3 5 , 4 0 , 4 5 , five 0 , v v , 6 0 , half dozen 5 , seven 0 , 7 5 , 8 0 . a n d
As the highest card value is 81, then there are no higher possible values. Counting these values, we encounter that at that place are 16.

Next, as there are 81 cards, so the total number of cards is 81.

Filling these values into the equation in a higher place gives
𝑃 ( 5 ) = one 6 viii 1 . d i 5 i s i b 50 e b y

Nosotros cannot simplify this fraction whatsoever further. Therefore, the probability of of picking a carte du jour with a number divisible by 5 from this deck of cards is

1 6 8 1 .

Nosotros volition now look at an case where we are given information about the total number of outcomes and the probability of an event to work out the number of outcomes of a specific consequence.

Example 4: Using Theoretical Probability to Solve a Trouble

At that place are 28 people in a meeting. The probability that a person called at random is a man is

ane 2 .
Summate the number of women in the coming together.

Respond

The question gives usa the value that the probability of choosing a man from the full number of people in the room is

1 2 .
We can use this information, forth with the information about the total number of people, to find the number of men in the room.

Recall that the probability of a simple event tin exist given as
p r o b a b i l i t y o f a north e v due east n t n u m b e r o f o u t c o m e s w h e r e t h east e v e due north t o c c u r due south t o t a l n u m b eastward r o f p o s s i b l e o u t c o yard e s = .

For this scenario, we tin can write the probability of picking a man,

𝑃 ( ) m a n ,
as
𝑃 ( ) = . m a n northward u m b due east r o f m eastward n i due north t h e yard e e t i n thou t o t a fifty n u m b e r o f p e o p l e i n t h east m e e t i northward 1000

Given that
𝑃 ( ) = 1 two k a n
and the

t o t a 50 n u 1000 b e r o f p e o p l e i due north t h e m e e t i n m = 2 8 ,
nosotros can substitute these values into the equation above, giving
i two = 2 8 . n u grand b east r o f m e n

Multiplying both sides of this equation by 28 and simplifying gives us
2 8 × 1 ii = ane 4 = . northward u chiliad b e r o f m e n northward u one thousand b e r o f yard due east n

Since the number of men in the meeting is 14, so nosotros can summate the number of women past subtracting the number of men, fourteen, from the total number of people, 28. Every bit
2 viii 1 4 = one 4 ,

the number of women in the coming together is xiv.

We will now look at an example where we find the probability of picking an even digit from a given number.

Example 5: Determining the Probability of an Result

If a single digit is selected at random from the number 224, 839, 287, what is the probability of the digit existence fifty-fifty?

Respond

Recall that the probability of a simple consequence is given by
p r o b a b i l i t y o f a n e v e n t n u m b east r o f o u t c o m e s westward h e r e t h e due east v e n t o c c u r s t o t a l due north u m b east r o f p o s s i b l e o u t c o m e due south = .

In this question, the probability of picking an even digit,

𝑃 ( ) e v e n d i thou i t ,
can exist constitute by
𝑃 ( ) = . e v due east north d i g i t n u 1000 b e r o f e five e northward d i 1000 i t south t o t a 50 due north u m b e r o f d i chiliad i t south

Considering each digit separately and determining its parity, we accept

Counting the number of even digits above, there are 6 even values. The total number of digits is nine. Substituting these values into the probability equation above gives united states of america
𝑃 ( ) = half dozen 9 = 2 iii . e five east n d i g i t

Therefore, the probability of selecting an even digit from the number 224, 839, 287 is

ii iii .

In the final instance, we will use a given probability and information nearly the specific number of an effect, to work out the total number of outcomes.

Example six: Using Theoretical Probability to Solve a Trouble

A purse contains 24 white balls and an unknown number of red balls. The probability of choosing a red ball at random is

seven 3 one .
How many balls are at that place in the purse?

Reply

Nosotros are given the number of white assurance in the purse, and we are given the probability of picking one of the unknown number of red balls as

7 three 1 .
To find the total number of balls, we can use the probability equation for finding the probability of a simple outcome:
p r o b a b i l i t y o f a north due east v eastward north t north u thousand b east r o f o u t c o m e s westward h east r east t h e due east 5 e n t o c c u r southward t o t a l n u m b e r o f p o southward southward i b l e o u t c o m due east due south = .

To discover the probability of getting a white ball, nosotros can recognize that since there are only red or white balls in the bag and the full of all of the probabilities is 1, and then the probability of getting white,

𝑃 ( ) westward h i t e ,
can be establish by
𝑃 ( ) = 1 𝑃 ( ) = 1 7 iii 1 = 2 4 3 i . w h i t due east r east d

The probability of picking a white ball can be plant past
𝑃 ( ) = . due west h i t e due north u m b eastward r o f w h i t eastward b a l fifty s t o t a l n u m b e r o f b a fifty fifty s

Substituting in the values
𝑃 ( ) = 2 four three 1 westward h i t e
and the number of white assurance = 24 gives us
2 four 3 one = 2 4 . t o t a l northward u m b e r o f b a l 50 s

Rearranging the equation past cross multiplying the fractions, we have
two iv × = iii 1 × ii 4 . t o t a fifty due north u m b e r o f b a l l s

Dividing both sides of the equation by 24 gives
t o t a l n u m b e r o f b a l fifty s = 3 i .

Nosotros tin check our answer by calculating that the number of ruby balls must exist

3 1 2 four = seven .
Thus, the probability of picking a cherry ball would be the given value of

vii three 1 ,
since
𝑃 ( ) = = 7 3 1 . r eastward d n u yard b due east r o f r e d b a l l due south t o t a fifty n u m b eastward r o f b a l 50 s

This confirms our reply; the full number of balls in the pocketbook is 31.

We now summarize the key points.

Key Points

  • The probability of an event is the likelihood of it occurring.
  • The sum of the probabilities of all possible outcomes must equal 1.
  • A probability experiment is considered as fair if all outcomes are every bit likely. Experiments that are unfair are often referred to as biased.
  • A simple event is an consequence with a unmarried outcome.
  • The probability of a simple event,

    𝑃 ( ) e v e northward t ,
    is
    𝑃 ( ) = . east v east n t n u m b eastward r o f o u t c o yard east southward w h e r e t h e due east v eastward n t o c c u r southward t o t a l n u m b e r o f p o south s i b l e o u t c o thousand east s

Which of the Following Describes a Simple Event

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