Use Excel to calculate probabilities for the following questions:

1. Suppose you conducted a study to find evidence for extra-sensory perception (ESP) by asking subjects to identify the color (red or black) of a card that is randomly selected from a deck of cards. You repeat this process 25 times and record the results.
   1. Given that there are an equal number of black and red cards in a deck of cards, what is the probability an individual can guess the color correctly?
   2. Would it be unusual for an individual to correctly identify the correct color 15 times? Justify your answer by calculating and interpreting the probability of correctly identifying the color 15 times in 25 trials.
   3. Would it be unusual for an individual to correctly identify the correct color only eight times? Justify your answer by calculating and interpreting the probability of correctly identifying the color 8 times in 25 trials.
   4. If an individual correctly identified the color correctly 23 times, would you consider that to be evidence of ESP? Justify your answer by calculating and interpreting the probability of correctly identifying the color 23 times in 25 trials.
   5. If an individual correctly identified the color correctly only 3 times, would you consider that to be evidence of ESP? Justify your answer by calculating and interpreting the probability of correctly identifying the color 3 times in 25 trials.
   6. Suppose 3 individuals from your study correctly identified the color 22 times. Assuming their results are independent of each other, what is the probability that all three correctly identified the 22 times? Would you consider this evidence that ESP is real?

(Hint: Calculate the probability of correctly identifying the color 22 times in 25 trials, and then use the assumption that the results are independent to calculate the probability of all 3 correctly identifying the color 22 times).

2. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

1. A friend of yours claims that her IQ is at least 120. Would you believe her claim? Justify your answer by calculating and interpreting the probability an individual has an IQ greater than 120
2. What is the probability that an individual has an unusually high IQ?(Hint: remember values that are more than 2 standard deviations above or below the mean are considered to be unusual).
3. If you randomly selected a professor and asked them for their IQ score, would you be surprised if their IQ was greater than 150? Justify your answer by calculating and interpreting the probability an individual has an IQ greater than 150.
4. Assume two professors claim to have IQs greater than 140. Would you believe their claim (Hint: calculate the probability an individual has an IQ greater than 140, and then assuming their scores are independent, calculate the probability both have an IQ greater than 140).
5. Is it correct to assume to calculate the probabilities for professors using the distribution for the general population? Justify your answer. Please show the calculation steps