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Here are a few recommended readings before getting started with this lesson.
In a multiple-choice test, Davontay randomly selected the answers to all five questions. Each question had two options to choose from.
Let $A$ be the event of guessing the answer to the second question correctly. Let $B$ be the event of guessing correctly on the fifth question. Are these events independent? If so, what does it mean for the events to be independent?To find out about the voting behavior of people according to their age, a survey was conducted outside of a polling station. The following table shows some data about the presidential election in $2016$ between Donald Trump and Hillary Clinton.
Clinton | Trump | |
---|---|---|
Young Adults $(18−29years old)$ | $55$ | $36$ |
Adults $(30−44years old)$ | $51$ | $41$ |
Middle-Aged Adults $(45−64years old)$ | $44$ | $52$ |
Seniors $(65+years old)$ | $45$ | $52$ |
Find the following conditional probabilities and describe their meaning in everyday words. Round each answer to two decimal places.
Descriptions:
Alternative description: If a person is selected at random among the surveyed young adults, there is a $60%$ chance they voted for Clinton and a $40%$ chance they voted for Trump.
Descriptions:
Alternative description: If a person that voted for Clinton is selected at random, there is a $26%$ chance that they are aged between $30$ and $44$ years old and a $23%$ chance that they are aged between $45$ and $64.$
Descriptions:
Descriptions:
Clinton | Trump | Total | |
---|---|---|---|
Young Adults $(18−29years old)$ | $55$ | $36$ | $91$ |
Adults $(30−44years old)$ | $51$ | $41$ | $92$ |
Middle-Aged Adults $(45−64years old)$ | $44$ | $52$ | $96$ |
Seniors $(65+years old)$ | $45$ | $52$ | $97$ |
Total | $195$ | $181$ | $376$ |
In other words, if a person is selected at random among the surveyed young adults, there is a $60%$ chance they voted for Clinton and a $40%$ chance they voted for Trump.
In other words, if a person that voted for Clinton is selected at random, there is a $26%$ probability that they are between $30$ and $44$ years old and a $23%$ probability that they are between $45$ and $64.$
Last month, Ignacio got a part-time job working from $4:00P.M.$ to $8:00P.M.$ during weekdays. Ignacio, who knows statistics, said to his peers that the events of $taking$ $a$ $nap$ $after$ $lunch$ and $being$ $late$ $for$ $work$ are independent. However, Tadeo, who does not know much about statistics, does not understand what Ignacio meant.
Late | On Time | Total | |
---|---|---|---|
Nap | $2$ | $6$ | $8$ |
No Nap | $3$ | $9$ | $12$ |
Total | $5$ | $15$ | $20$ |
The events of $taking$ $a$ $nap$ $after$ $lunch$ and $being$ $late$ $for$ $work$ are independent.
Ignacio is saying that the probability that he is late for work is the same whether or not he takes a nap after lunch. Therefore, on the days that Ignacio is late for work, the nap is not the cause. With this explanation, Tadeo will hopefully understand what Ignacio meant.
Ignacio taking a nap after lunchand
Ignacio being late for workare independent using the data from the table. Remember, if two events $A$ and $B$ are independent, then $P(A)$ is equal to $P(A∣B).$ Therefore, the following equation needs to be checked.
In Maya's new neighborhood, some people have dogs, cats, both, or neither. The following diagram shows the distribution of pets, but Maya has not seen it.
Notice that as they are written, these probabilities represent the same situation. However, Maya knows that Magdalena does not have a cat and that Dylan does not have a dog. With this information, the above probabilities can be rewritten.
These are conditional probabilities. Since there are only two types of pets in the survey, the above statement can be written more precisely.
By the Complement Rule, the following pair of conclusions can also be drawn.
From the four statements, Maya could safely ask Magdalena about her dog, but she should not ask Dylan about his cat. Keep in mind that Magdalena might not have a dog despite the probabilities and conclusions. Similarly, Dylan might have a cat.
Tearrik wants to determine if there is a connection between age and music preference. To figure it out, he surveyed $120$ people at the mall, asking their age and whether they prefer pop or classical music. After analyzing the data collected, he concluded that there is no connection at all.
Pop | Classical | Total | |
---|---|---|---|
$35$ Years Old or Younger | |||
Older Than $35$ | $40$ | ||
Total | $45$ | $120$ |
Based on the conclusion made by Tearrik, complete the missing information in the two-way frequency table.
Pop | Classical | Total | |
---|---|---|---|
$35$ Years Old or Younger | $30$ | $50$ | $80$ |
Older Than $35$ | $15$ | $25$ | $40$ |
Total | $45$ | $75$ | $120$ |
Since there is no connection between age and music preference, the probability that someone older than $35$ likes pop music is the same as the probability that any person likes this type of music. In other words, the events A person likes pop music
and A person is older than $35$ years old
are independent.
For simplicity, some variables will be assigned to the missing data.
Pop | Classical | Total | |
---|---|---|---|
$35$ Years Old or Younger | $a$ | $b$ | $c$ |
Older Than $35$ | $d$ | $e$ | $40$ |
Total | $45$ | $f$ | $120$ |
totalrow and column add to the grand total $120,$ the missing marginal frequencies can be calculated.
Pop | Classical | Total | |
---|---|---|---|
$35$ Years Old or Younger | $a$ | $b$ | $80$ |
Older Than $35$ | $d$ | $e$ | $40$ |
Total | $45$ | $75$ | $120$ |
To find the joint frequencies, the conclusion made by Tearrik will be used instead of a system of equations.
There is no connection between age and music preference.
$P(Pop∣>35)=40d $, $P(Pop)=83 $
$LHS⋅40=RHS⋅40$
$ca ⋅b=ca⋅b $
Calculate quotient
Pop | Classical | Total | |
---|---|---|---|
$35$ Years Old or Younger | $a$ | $b$ | $80$ |
Older Than $35$ | $15$ | $e$ | $40$ |
Total | $45$ | $75$ | $120$ |
Pop | Classical | Total | |
---|---|---|---|
$35$ Years Old or Younger | $30$ | $b$ | $80$ |
Older Than $35$ | $15$ | $25$ | $40$ |
Total | $45$ | $75$ | $120$ |
Pop | Classical | Total | |
---|---|---|---|
$35$ Years Old or Younger | $30$ | $50$ | $80$ |
Older Than $35$ | $15$ | $25$ | $40$ |
Total | $45$ | $75$ | $120$ |
Note that the conclusion made by Tearrik implies that the following pairs of events are independent.
A person likes pop musicand
A person is older than $35$ years old.
A person likes pop musicand
A person is $35$ years old or younger.
A person likes classical musicand
A person is older than $35$ years old.
A person likes classical musicand
A person is $35$ years old or younger.
Diego wants to throw a party at the end of the school year. To determine what kind of treats he should buy, he asked his $80$ classmates whether they prefer cupcakes, cookies, donuts, or chocolate.
On the day of the party, Diego puts the treats on a table.
Substitute values
Calculate quotient
Convert to percent
Round to nearest integer
Substitute values
Calculate quotient
Convert to percent
Round to nearest integer
Substitute values
Calculate quotient
Convert to percent
Round to nearest integer
Substitute values
Calculate quotient
Convert to percent
Round to nearest integer
Substitute values
Calculate quotient
Convert to percent
Round to nearest integer
Substitute values
$ba =b/16a/16 $
Substitute values
$ba =b/9a/9 $
Substitute values
$ba =b/7a/7 $
Davontay took a multiple-choice test where each question had two choices. He randomly guessed the answers to all the five questions in the test.
Let $A$ be the event of guessing the answer to the second question correctly. Let $B$ be the event of guessing correctly on the fifth question.
Substitute values
Multiply fractions
The events being independent means that having guessed the second question correctly does not influence having guessed the fifth question correctly and vice versa.