# Researchers carried out a survey of fourth-, fifth- and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data. begin{array}{c|c}& 4th grade & 5th grade & 6th grade &Total hline Grades &49&50&69&168 text{Athletic} &24&36&38&98 text{Popular} &19&22&28&69 hline text{Total} & 92 & 108 & 135 &335 end{array} Suppose we select one of these students at random. What's the probability of each of the following? The student is a sixth-grader or rated good grades as Important.

Question
Two-way tables
Researchers carried out a survey of fourth-, fifth- and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.
$$\begin{array}{c|c}& 4th\ grade & 5th\ grade & 6th\ grade &Total \\ \hline Grades &49&50&69&168\\ \text{Athletic} &24&36&38&98\\ \text{Popular}\ &19&22&28&69\\ \hline \text{Total} & 92 & 108 & 135 &335 \end{array}$$
Suppose we select one of these students at random. What's the probability of each of the following? The student is a sixth-grader or rated good grades as Important.

2021-01-28
Given:
$$\begin{array}{c|c}& 4th\ grade & 5th\ grade & 6th\ grade &Total \\ \hline Grades &49&50&69&168\\ \text{Athletic} &24&36&38&98\\ \text{Popular}\ &19&22&28&69\\ \hline \text{Total} & 92 & 108 & 135 &335 \end{array}$$
The table contains data of 335 students, which is given in the bottom right corner of the given table.
$$\text{# of possible outcomes} = 335$$
All students who are 6th grades and rated good grades as important are given in the row "6th grade” or in the column ”Grades” of the given table. Adding all corresponding counts, we then note that this corresponds with 49+50+69+38 +28=211 students who are 6th graders or rated good grades as important.
$$\text{# of favorable outcomes} = 234$$
The probability is the number of favorable outcomes divided by the number of possible outcomes:
$$P(\text{ 6th grade or Grades})=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}$$
$$=\frac{234}{335}$$
$$\approx 0.6985$$
$$=69.85\%$$

### Relevant Questions

Researchers carried out a survey of fourth-, fifth- and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.
$$\begin{array}{c|c} & 4th\ grade & 5th\ grade & 6th\ grade &Total \\ \hline Grades &49&50&69&168\\ Athletic &24&36&38&98\\ Popular\ &19&22&28&69\\ \hline Total & 92 & 108 & 135 &335 \end{array}$$
Suppose we select one of these students at random. What's the probability of each of the following? The student is not a sixth-grader and did not rate good grades as important.

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. This two-way table summarizes the survey data.

$$\begin{array} {|c|} & Grade \ Most important & \begin{array}{l|c|c|c|c} & \begin{array}{c} 4 \mathrm{th} \ grade \end{array} & \begin{array}{c} 5 \mathrm{th} \ \text { grade } \end{array} & \begin{array}{c} 6 \mathrm{th} \ grade \end{array} & Total \\ \hline Grades & 49 & 50 & 69 & 168 \\ \hline Athletic & 24 & 36 & 38 & 98 \\ \hline Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array} \ \end{array}$$

Suppose we select one of these students at random. What's the probability that: The student is a sixth grader or a student who rated good grades as important?

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.

$$Grade\ Most\ important\begin{array}{l|c|c|c|c} & 4 \mathrm{th} & 5 \mathrm{th} & 6 \mathrm{th} & \text { Total } \\ \hline Grades & 49 & 50 & 69 & 168 \\ \hline Athletic & 24 & 36 & 38 & 98 \\ \hline Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array}$$

Suppose we select one of these students at random. Find P(athletic | 5th grade).

The Pew Research Center asked a random sample of 2024 adult cellphone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.
$$\begin{array}{c|ccc|c} & 18-34 & 35-54 & 55+ & \text { Total } \\ \hline \text { iPhone } & 169 & 171 & 127 & 467 \\ \text { Androod } & 214 & 189 & 100 & 503 \\ \text { Other } & 134 & 277 & 643 & 1054 \\ \hline \text { Total } & 517 & 637 & 870 & 2024 \end{array}$$
Suppose we select one of the survey respondents at random. What's the probability that: The person is not age 18 to 34 and does not own an iPhone?
Is there a relationship between gender and relative finger length? To find out, we randomly selected 452 U.S. high school students who completed a survey. The two-way table summarizes the relationship between gender and which finger was longer on the left hand (index finger or ring finger).
$$\begin{array} {lc} & \text{Gender} \ \text {Longer finger} & \begin{array}{l|c|r|r} & \text { Female } & \text { Male } & \text { Total } \\\hline \text { Index finger } & 78 & 45 & 123 \\\hline \text{ Ring finger } & 82 & 152 & 234 \\ \hline \text { Same length } & 52 & 43 & 95 \\ \hline \text { Total } & 212 & 240 & 452 \end{array}\ \end{array}$$
Suppose we randomly select one of the survey respondents. Define events R: ring finger longer and F: female. Given that the chosen student does not have a longer ring finger, what's the probability that this person is male? Write your answer as a probability statement using correct symbols for the events.
A survey of 4826 randomly selected young adults (aged 19 to 25 ) asked, "What do you think are the chances you will have much more than a middle-class income at age 30? The two-way table summarizes the responses.
$$\begin{array} {c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but probably not } & 426 & 286 & 712 \\ \hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}$$
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Given that the chosen student didn't say "almost no chance," what's the probability that this person is female? Write your answer as a probability statement using correct symbols for the events.
A group of 125 truck owners were asked what brand of truck they owned and whether or not the truck has four-wheel drive. The results are summarized in the two-way table below. Suppose we randomly select one of these truck owners.
$$\begin{array}{c|cc} & \text { Four-wheel drive} & \text { No four-wheel drive } \\\hline \text { Ford } & 28 & 17 \\ \text { Chevy } & 32 & 18 \\ \text { Dodge } & 20 & 10 \end{array}$$
What is the probability that the person owns a Dodge or has four-wheel drive?
$$(a) \frac{20}{80}$$
$$(b) \frac{20}{125}$$
$$(c) \frac{80}{125}$$
$$(d) \frac{90}{125}$$
$$(e) \frac{110}{125}$$
1950 randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two-way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same as, or worse off than their parents
$$\begin{array}{|c|c|c|}\hline &\text{Less Than High School}&\text{High School}&\text{More Than High School}\\\hline \text{Better off} &140&440&430\\ \hline \text{Same as}&60&230&110\\ \hline \text{Worse off}&180&280&80\\ \hline\end{array}\\$$
Suppose one adult is selected at random from these 1950 adults. Find the following probablity.
$$P(\text{more than high school or worse off})=?$$
$$\begin{array}{c|c} & Female\ \ \ \ \ \ Male & Total \\ \hline About\ right & 560\ \ \ \ \ \ \ \ \ \ 295 & 855\\ \hline Overweight & 163\ \ \ \ \ \ \ \ \ \ 72 & 235 \\ \hline Underweight & 37\ \ \ \ \ \ \ \ \ \ \ \ 73 & 110 \\ \hline Total & 760\ \ \ \ \ \ \ \ \ \ 440 & 1200 \end{array}$$
$$\begin{array}{c|c} & First & Second & Third \\ \hline Yes & 197 & 94 & 151\\ \hline No & 122 & 167 & 476\\ \end{array}$$