The two-way table shows the classification of students in a mathematics class by gender and dominant hand. A student who is ambidextrous uses both hands equally well.

Question

right hand left hand table

One student will be selected at random from the class. Consider events X and Y shown below.

X: the selected student is female
Y: the selected student is right-handed
Which of the following statements about events X and Y is TRUE?

Question 3 options:

The events are not independent because the probability of X is not equal to the probability of X given Y.

The events are independent because the number of categories for dominant hand is different from the number of categories for gender.

The events are independent because the number of right-handed students in the class is larger than the number of female students.

The events are not independent because for one of the dominant hand categories the number of female students is 0.

Answer 1

To determine whether events X and Y are independent, we must first understand what it means for two events to be independent. Events X (the selected student is female) and Y (the selected student is right-handed) are independent if the probability of X occurring does not change given that Y has occurred. Mathematically, this is represented as:

\[ P(X \mid Y) = P(X) \]

Now, let's analyze the options provided:

1. The events are not independent because the probability of X is not equal to the probability of X given Y.
   This statement is generally true if we find that the probabilities do not equal each other. If in practice we calculate \( P(X \mid Y) \) and it's not equal to \( P(X) \), this would be a correct statement.

2. The events are independent because the number of categories for dominant hand is different from the number of categories for gender.
   The number of categories does not determine independence. Independence is based on probabilities, not the number of categories.

3. The events are independent because the number of right-handed students in the class is larger than the number of female students.
   Again, independence does not depend on the size of categories but rather on the relationship of the probabilities.

4. The events are not independent because for one of the dominant hand categories the number of female students is 0.
   This could imply some relationship and would not satisfy independence if there are no females in one of the right-handed categories.

In the absence of the actual counts of female and right-handed students from the provided data, the fourth option seems most indicative of potential dependence, as it suggests that the gender distribution might influence the hand dominance distribution.

If you only could choose one option based on reasoning alone without specific data, the last option could likely be the correct answer since it explicitly indicates a relationship where one category has no representation, which typically suggests that the events may depend on each other.

Thus, the true statement among the given options is:

The events are not independent because for one of the dominant hand categories the number of female students is 0.