Final answer:
The correct representation of the random variable for counting the total number of tails in 75 flips of a weighted coin where tails are six times as likely as heads is B(75, 6/7). The answer choice that most closely represents this is option E, B(75, 67), assuming some rounding to the nearest integer for the probability.
Step-by-step explanation:
The student asked to identify which random variable would be used to count the total number of tails flipped when we have a weighted coin that flips tails six times as likely as heads. The experiment involves flipping the coin seventy-five times.
The correct way to represent this situation would be with a binomial random variable, notated as B(n, p), where 'n' represents the number of trials (coin flips) and 'p' represents the probability of success on a single trial (in this case, landing a tail).
Since the coin is weighted in such a way that tails are six times as likely as heads, if we let the probability of flipping heads be 'x', then the probability of flipping tails would be '6x'. Because there can only be two outcomes and all probabilities must add up to 1, we have x + 6x = 1, thus 7x = 1, or x = 1/7. Therefore, the probability of flipping tails is 6/7.
With 75 flips, the random variable to represent the total number of tails would be B(75, 6/7). We need to convert the fraction to its closest decimal to find the right option. The decimal of 6/7 is approximately 0.857, which isn't directly reflected in any of the answer choices. However, the choice that comes closest to the actual probability we calculated (with rounding) and the number of flips (75) is option E, B(75, 67), if we consider 67 is closest to 75*(6/7).