# 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 th

Question
Two-way tables
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.

2020-10-27
DEFINITIONS
Definition conditional probability:
$$P(B|A)=\frac{P(A \cap B)}{P(A)}=\frac{P(A\ and\ B)}{P(A)}$$
SOLUTION
$$\begin{array} {lc} & \text{Gender} \ \text {Longer finger} & \begin{array}{l|c|r|r} & \text { Female } & \text { Male } & \text { Total } \\\hline \text { Index finger } & 78 & 45 & 123 \\\hline \text{ Ring finger } & 82 & 152 & 234 \\ \hline \text { Same length } & 52 & 43 & 95 \\ \hline \text { Total } & 212 & 240 & 452 \end{array}\ \end{array}$$
R=ring finger longer
F=female
We note that the table contains information about 452 students (given in the bottom right corner of the table).
Moreover, 123+95=218 of the 452 students are female, because 123 and 95 are mentioned in the row "Index Finger/Same Length" and in the column "Total" of the table.
The probability is the number of favorable outcomes divided by the number of possible outcomes:
$$P(R^c)=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}=\frac{218}{452}$$
Next, we note that 45+43=88 of the 452 students are males that don't have a longer ring finger, because 45 and 43 are mentioned in the row "Index finger/Same Length" and in the column "Male" of the given table.
$$P(R^c\ and\ F^c)=\frac{\text{# of favorable outcomes}}{\text{# of possible outcomes}}=\frac{88}{452}$$
Use the definition of conditional probability:
$$P(F^c|R^c)=\frac{P(R^c\ and\ F^c)}{P(R^c)}=\frac{88/452}{218/452}=\frac{88}{218}=\frac{44}{109}\approx0.4037=40.37\%$$
40.37% of the people with no longer right finger are males.
Result:
$$\frac{44}{109}\approx0.4037=40.37\%$$
40.37% of the people with no longer right finger are males.

### Relevant Questions

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). $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{c}\right\rbrace}&\text{Gender}\backslash\text{Longer finger}&{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}{\left|{c}\right|}{r}{\mid}{r}\right\rbrace}&\ \text{ Female }\ &\ \text{ Male }\ &\ \text{ Total }\ \backslash{h}{l}\in{e}\ \text{ Index finger }\ &{78}&{45}&{123}\backslash{h}{l}\in{e}\ \text{ Ring finger }\ &{82}&{152}&{234}\backslash{h}{l}\in{e}\ \text{ Same length }\ &{52}&{43}&{95}\backslash{h}{l}\in{e}\ \text{ Total }\ &{212}&{240}&{452}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\backslash{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Suppose we randomly select one of the survey respondents. Define events R: ring finger longer and F: female. Find P(R|F). Interpret this value in context.
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.
$$\begin{array} {c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but probably not } & 426 & 286 & 712 \\ \hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}$$
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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. PSK\begin{array} {lc} & 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}ZSK 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?
A random sample of 88 U.S. 11th- and 12th-graders was selected. The two-way table summarizes the gender of the students and their response to the question "Do you have allergies?" Suppose we choose a student from this group at random.
$$\begin{array}{c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text{ Yes } & 19 & 15 & 34 \\ \text{ No } & 24 & 30 & 54 \\ \hline \text{ Total } & 43 & 45 & 88\\ \end{array}\$$
What is the probability that the student is female or has allergies?
$$(a)\frac{19}{88}$$
(b)\frac{39}{88}\)
(c)\frac{58}{88}\)
(d)\frac{77}{88}\)
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. PSK\begin{array} {lc} & \text{Gender} \ \text {Opinion} & \begin{array}{l|c|c|c} & Female & Male & Total \\ \hline Almost no chance & 96 & 98 & 194 \\ \hline \begin{array}{l} Some chance but \\ robably not \end{array} & 426 & 286 & 712 \\ \hline A 50-50 chance & 696 & 720 & 1416 \\ \hline A good chance & 663 & 758 & 1421 \\ \hline Almost certain & 486 & 597 & 1083 \\ \hline Total & 2367 & 2459 & 4826 \end{array}\ \end{array}ZSK Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Find P(G | M). Interpret this value in context.
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$$\begin{array}{c|ccc|c} & 18-34 & 35-54 & 55+ & \text { Total } \\ \hline \text { iPhone } & 169 & 171 & 127 & 467 \\ \text { Androod } & 214 & 189 & 100 & 503 \\ \text { Other } & 134 & 277 & 643 & 1054 \\ \hline \text { Total } & 517 & 637 & 870 & 2024 \end{array}$$
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?
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.
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.
$$\begin{array}{ccc}\text{Original Counts}&\text{Exciting}&\text{Routine}&\text{Dull}&\text{Total}\\\hline \text{Male} &213&200&12&425\\ \text{Female}&221&305&29&555\\ \text{Female}&434&505&41&980 \end{array}$$
$$\begin{array} {lc} & \text{Class} \ \text {Survived } & \begin{array}{c|c|c|c} & \text { First } & \text { Second } & \text { Third } \\ \hline \text { Yes } & 197 & 94 & 151 \\ \hline \text { No } & 122 & 167 & 476 \end{array}\ \end{array}$$