1. According to Wikipedia, the probability of a Royal Flush in one hand of poker is 0.000154% (note the percent sign). If 10,000 hands are played at a casino each night, what is the probability that there will be at least one Royal Flush in any given night, assuming no cheating and the deck is well shuffled (resuffled) for each hand?
  1. If, as my Baseball research suggests, 2% of half-innings are “new” half innings, what is the probability that an 18 half-inning game will have at least one “new” half inning, assuming independence of half-innings?

  1. According to the Wikipedia, Paul, the famous Octopus from the 2010 World Cup, chose the winning team 12 out of 14 times. If each selection had a probability of 1/2, and if each selection was independent, (together the Null hypothesis), what is the probability (p-value) that Paul would have chosen 12 or more times correctly?

  2. Does this result suggest that either Paul has special powers, or that the assumptions are incorrect? (You don’t have to choose between these possibilities, see Wikipedia for a discussion).

  1. A soothsayer claims to have exta-sensory perception and asserts that she can predict the outcomes of coin tosses. You want to test her claim with a hypothesis test, using the Binomial calculator. Assume each outcome of the coin (heads or tails) is equally likely, and 100 tosses are equally likely. Let the null hypothesis be that her selection has a 50-50 chance of being right. If she gets 60 tosses correct, what is the probability she will get 60 or more tosses correct?

  2. Would her claim stand up to scrutiny with a hypothesis test if the level of significance of the test were 0.05?

  1. A four sided die is thrown and the sides are labeled with colors: red, green, blue, and white. Give an example of a random variable on this sample space.

  2. A random student is chosen from our class. A question is asked of the student, and based on the answer to this question, a random variable is defined. What is one possibility for this question?

  3. Three random students are chosen from our class. A question is asked of each student, and based on the answers to this question, a random variable is defined. What is one possibility for this question, and how is the random variable defined.

  4. Is a random variable random?

  5. Is a random variable a variable?

  6. What is a random variable?

Answers

  1. 0.0153 (equivalent to 1.53% of nights).

  2. 0.301 (equivalent to 30.1% of games).

  3. 0.00647.

  4. Yes, one or the other, or Paul was very lucky.

  5. 0.0443.

  6. Yes.

  7. Answers vary. Discussed in class.

  8. Answers vary. Discussed in class.

  9. Answers vary. Discussed in class.

  10. NO!

  11. NO!

  12. A random variable is a function that assigns a number to each outcome of a random phenomenon.