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?
