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Since the stock market began in 18721872, stock prices have risen in about 73%73\% of the years. Assuming that market performance is independent from year to year, what's the probability that the market will fall during at least 11 of the next 55 years?

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Q. Since the stock market began in 18721872, stock prices have risen in about 73%73\% of the years. Assuming that market performance is independent from year to year, what's the probability that the market will fall during at least 11 of the next 55 years?
  1. Calculate Probability of Market Rise: First, we need to determine the probability that the stock market rises in any given year, which is given as 73%73\%. To find the probability that the market falls in any given year, we subtract this probability from 100%100\%.\newlineCalculation: 100%−73%=27%100\% - 73\% = 27\%
  2. Calculate Probability of Market Not Falling for 55 Years: Next, we need to calculate the probability that the market will not fall for 55 consecutive years. Since the performance of the market is independent from year to year, we can multiply the probability of the market not falling in a single year by itself five times.\newlineCalculation: (1−0.27)5(1 - 0.27)^5
  3. Evaluate Probability of Market Not Falling for 55 Years: Now, we evaluate the expression from the previous step to find the probability that the market will not fall for 55 consecutive years.\newlineCalculation: (0.73)5≈0.1849(0.73)^5 \approx 0.1849
  4. Find Probability of Market Falling in Next 55 Years: To find the probability that the market will fall during at least 11 of the next 55 years, we subtract the probability that the market will not fall for 55 consecutive years from 11. Calculation: 1−0.1849≈0.81511 - 0.1849 \approx 0.8151

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