Question

In 42 of the 64 years from 1950 through 2014,the S& P 500 finished higher after the first five days of trading (in 2011 there was virtually no change). In 37 out of 42 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 64-year period.

S&P 500's Annual Performance
First week Higher Lower Total
Higher 37 5 42
Lower 11 11 22
Total 48 16 64

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

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

c) Are the two events "first week performance" and "annual performance"
independent or dependent? Explain why.

Answer 1

The probability that the S&P 500 finishes lower in a randomly selected year is 25%. If the S&P 500 finishes lower after the first week, the probability it finishes lower for the year is 50%. The first-week performance and annual performance appear to be dependent events.

Step-by-step explanation:

The probability of the S&P 500 finishing lower for a year selected at random is calculated by dividing the number of years it finished lower (16) by the total number of years (64), which gives a probability of 0.25 or 25%. This answers part (a).

For part (b), the probability that the S&P 500 finished lower for the year given that it finished lower after the first five days of trading is calculated by dividing the number of years it finished lower given a lower first week (11) by the total number of years it finished lower after the first week (22), resulting in a probability of 0.5 or 50%.

To determine if the two events are independent or dependent for part (c), we can check whether the probability of the annual performance being higher or lower is the same regardless of the first week's performance. Since 37 out of 42 is significantly different from 11 out of 22, these events seem to be dependent, suggesting that there might be a correlation between a 'good' first week and a higher finishing for the year.

Answer 2

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