In a class there are four freshman boys, six freshman girls, and six sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?
Imagine a class of 20 students, composed of four freshman boys, six freshman girls, and six sophomore boys. The question is, how many sophomore girls must be present in the class if sex (boy or girl) and class (freshman or sophomore) are to be independent when a student is selected at random?
In order for class and gender to be independent, the probability of a student being a freshman boy and the probability of a student being a boy must be the same. That is, the probability of being a freshman boy must be equal to the probability of being a boy overall. The same holds true for being a sophomore girl.
The probability of being a freshman boy is 4/20 and the probability of being a boy is 10/20. To make these equal, we need to add more sophomore girls such that the probability of being a freshman boy is equal to the probability of being a boy overall.
This can be represented by the equation: 4/20 = (4 + x)/(20 + x), where x represents the number of sophomore girls.
Solving for x, we get x = 6.
Therefore, there must be 6 sophomore girls in the class in order for class and gender to be independent when a student is selected at random.
In summary, when considering the independence of class and gender in a class of students, it is important to ensure that the probabilities of each combination of class and gender are equal. In this scenario, the presence of 6 sophomore girls in the class of 20 students satisfied this condition.
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