Historically, the likelihood for a stock going up in a particular year is 53% - a little better than a coin flip. According to Investopedia, there seems to be a January Effect in the stock market. If there is a January rally, then the probability that the market will have a good year is 60%. Statistics also show that the probability of a January rally is 62%.
(a) What is the probability that there is a January rally followed by a good year?
(b) If the yearly performance is in positive territory, what is the probability that there was no January rally?
(c) What is the probability of encountering a bear market in both January and the whole year?
(d) Are the two events "January rally" and "Good year" independent? Mutually exclusive? Why?