Question

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).
suppose we randomly select one of the survey respondents. define events r: ring finger longer and f: female.
a) find p(r | f). interpret this value in context.
b) given that the chosen student does not have a longer ring finger, whats the probability that this person is male? write your answer as a probability statement using correct symbols for the events.

Explanation:

Step1: Recall conditional probability formula

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{n(A\cap B)}{n(B)}\)

Step2: Find \(n(R\cap F)\) and \(n(F)\)

From the table, \(n(R\cap F) = 82\) (number of females with ring - finger longer) and \(n(F)=212\) (total number of females)

Step3: Calculate \(P(R|F)\)

\(P(R|F)=\frac{82}{212}=\frac{41}{106}\approx0.387\)

Step4: Interpret \(P(R|F)\)

If we randomly select a female from the survey respondents, there is approximately a \(38.7\%\) chance that her ring finger is longer.

Step5: For part b)

First, find the number of students who do not have a longer ring finger. \(n(\text{not }R)=452 - 234=218\)
The number of males who do not have a longer ring finger: \(n(\text{not }R\cap M)=45 + 43=88\)
Using the conditional - probability formula \(P(M|\text{not }R)=\frac{n(\text{not }R\cap M)}{n(\text{not }R)}\)
\(P(M|\text{not }R)=\frac{88}{218}=\frac{44}{109}\approx0.404\)

Answer:

a) \(P(R|F)=\frac{41}{106}\approx0.387\). Interpretation: If we randomly select a female from the survey respondents, there is approximately a \(38.7\%\) chance that her ring finger is longer.

b) \(P(M|\text{not }R)=\frac{44}{109}\approx0.404\)