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
asked 2020-10-26
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.

Answers (1)

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.
0

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