In 1964, two dolphins (Buzz and Doris) were trained to push one of two buttons in a pool in reaction to a light. If the light flashed, the dolphins pushed the button on the left to get a fish. If the light was constant, the dolphins needed to push the button on the right to get a fish. Suppose the researchers are interested in whether Buzz has truly learned the behavior to push a button correctly. They randomly make the light shine constant or flash 20 times, and record whether Buzz pushes the correct button. Define the random variable X as the number of times in which Buzz pushes the correct button out of 20 attempts. If Buzz has not learned the behavior, he would push buttons randomly and the probability that Buzz pushes the correct button is 0.50.

5a.) Using the appropriate distribution of X, explain why you would not be impressed if Buzz pushed the correct button 10 (or 9 or 11 times)?

5b.)How many successful attempts (out of 20) would it take for you to suggest that Buzz, perhaps, has learned the behavior to push a button correctly? Explain your choice.

5c.)Suppose now we increase the number of attempts to 200. Explain how this increase in attempts would affect the distribution of X?