The random walk model of stock prices states that stock market returns are independent of the returns in other periods; for example, whether the stock market does well or poorly in the coming month does not depend on whether it has done well or poorly during the past month, the past 12 months, or the past 120 months. On average, the monthly return on US stocks has been positive about 60 percent of the time and negative about 40 percent of the time. If monthly stock returns are independent with a 0.6 probability of a positive return and a 0.4 probability of a negative return, what is the probability of:

    a. 12 consecutive positive returns?

    b. 12 consecutive negative returns?

    c. A positive return, if the return the preceding month was negative?