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?
Independence between two events means that the occurrence of one event does not affect the probability of the other event. In this scenario, the events are the class (freshman or sophomore) and gender (boy or girl) of a student selected at random.
To determine if class and gender are independent, we must calculate the probability of selecting a freshman and a girl, and compare it to the probability of selecting a freshman multiplied by the probability of selecting a girl.
The total number of students in the class is 4 freshman boys + 6 freshman girls + 6 sophomore boys = 16 students. The probability of selecting a freshman is (4 + 6)/16 = 10/16. The probability of selecting a girl is (6 + 0)/16 = 6/16.
The probability of selecting a freshman and a girl is 6/16. The probability of selecting a freshman multiplied by the probability of selecting a girl is (10/16) * (6/16) = 0.375.
Since the probability of selecting a freshman and a girl is not equal to the probability of selecting a freshman multiplied by the probability of selecting a girl, we can conclude that class and gender are not independent.
To make class and gender independent, we need to ensure that the probability of selecting a freshman and a girl is equal to the probability of selecting a freshman multiplied by the probability of selecting a girl. Hence, the number of sophomore girls must be 4 to make the total number of students in the class equal to 20.
We have determined that class and gender are not independent when a student is selected at random in a class with 4 freshman boys, 6 freshman girls, and 6 sophomore boys. To make class and gender independent, we must have 4 sophomore girls in the class.
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