Question 5.84

84. Paul the Predicting Octopus. During the 2010 World Cup, Paul the Octopus picked the correct outcome of each soccer match in which Germany was involved, and then he continued his winning streak right to the end, picking champion Spain in the final match. Paul indicated his choice by swimming over and choosing one of two containers of mussels (his favorite food). Each container had the flag of a country in the day's match. Paul predicted the outcome correctly eight times in a row.

  1. Find the probability of Paul predicting all eight matches correctly, assuming a success probability of 50% for each match. Express this answer in the form , where is the number of matches.
  2. Let's say that Paul had predicted nine matches in a row correctly. Find the probability of doing so, assuming 50% probability of success for each match. Express this answer in the form .
  3. What is the ratio of your results from (b) to your results from (a)?
  4. Use your results from (a)–(c) to form a general rule for the probability of correctly predicting matches in a row when the probability of success for each match is 50%.