In 44 of the 68 years from 1950 through 2018 (in 2011 there was virtually no change), the S&P 500 finished higher after the first five days of trading. In 36 out of 44 years, the S&P 500 finished higher for the year. Is a good first week a good omen for the upcoming year? The following table gives the first-week and annual performance over this 68-year period:

a. If a year is selected at random, what is the probability that the S&P 500 finished higher for the year?

b. Given that the S&P 500 finished higher after the first five days of trading, what is the probability that it finished higher for the year?

c. Are the two events "first-week performance" and "annual performance" independent? Explain.

d. Look up the performance after the first five days of 2019 and the 2019 annual performance of the S&P 500 at finance.yahoo.com. Comment on the results.

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S&P 500'S ANNUAL PERFORMANCE FIRST WEEK Higher Lower Higher 36 8 Lower 12 12