The Standard and Poor 500 (S&P 500) is a weighted average of the stocks for 500 large companies in the United States. It is commonly used as a measure of the overall performance of the US stock market. Between January 1, 2009 and January 1, 2012, the S&P 500 increased for 423 of the 756 days that the stock market was open. We will investigate whether changes to the S&P 500 are independent from day to day. This is important, because if changes are not independent, we should be able to use the performance on the current day to help predict performance on the next day.

(a) What is the probability that the S&P 500 increased on a randomly selected market day between January 1, 2009 and January 1, 2012?

(b) If we assume that daily changes to the S&P 500 are independent, what is the probability that the S&P 500 increases for two consecutive days? What is the probability that the S&P 500 increases on a day, given that it increased the day before?

(c) Between January 1, 2009 and January 1, 2012 the S&P 500 increased on two consecutive market days 234 times out of a possible 755. Based on this information, what is the probability that the S&P 500 increases for two consecutive days? What is the probability that the S&P 500 increases on a day, given that it increased the day before?

(d) Compare your answers to part (b) and part (c). Do you think that this analysis proves that daily changes to the S&P 500 are not independent?