Suppose you believe that the price of a particular stock goes up each day with probability p and goes down with probability 1 - p. You also believe the daily price changes are independent of one another.

However, you are not sure of the value of p. Based on your current information, you believe p could be 0.40, 0.45, 0.50, or 0.55, with probabilities 0.15, 0.25, 0.35, and 0.25, respectively.

Then you watch the stock price changes for 25 days and observe 12 ups and 13 downs.

Use Bayes' rule to find the posterior distribution of p.

Based on this posterior distribution, calculate the probability that there will be at least 15 ups in the next 30 price changes. (Hint: Think in terms of the binomial distribution.)