1  Randomness and Probability

Probability comes up in a wide variety of situations. Consider just a few examples.

  1. The probability you win the next Powerball lottery if you purchase a single ticket, 4-8-15-16-42, plus the Powerball number, 23.
  2. The probability that a “randomly selected” Cal Poly student is a California resident.
  3. The probability that the high temperature in San Luis Obispo next Friday is above 90 degrees F.
  4. The probability that the San Francisco 49ers win the next Superbowl.
  5. The probability that extraterrestrial life currently exists somewhere in the universe.
  6. The probability that you ate an apple on April 17, 2009.

Example 1.1

How are the situations above similar, and how are they different? What is one feature that all of the situations have in common? Is the interpretation of “probability” the same in all situations? The goal here is to just think about these questions, and not to compute any probabilities (or to even think about how you would).





Example 1.2

One of the oldest documented problems in probability is the following: If three fair six-sided dice are rolled, what is more likely: a sum of 9 or a sum of 10?

  1. Explain how you could conduct a simulation to investigate this question.




  2. Use the simulation results to approximate the probability that the sum is 9; repeat for a sum of 10.




  3. It can be shown that the theoretical probability that the sum is 9 is 25/216 = 0.116. Write a clearly worded sentence interpreting this probability as a long run relative frequency.




  4. It can be shown that the theoretical probability that the sum is 10 is 27/216 = 0.125. How many times more likely is a sum of 10 than a sum of 9?




# number of repetitions; each rep is set of three rolls
N_reps = 10000

# vector to store results
sum_of_rolls = vector(length = N_reps) 

# for loop for the simulation
for (i in 1:N_reps) {
  
  # roll the dice
  rolls = sample(1:6, size = 3, replace = TRUE)
  
  # compute the sum of the rolls, and store the value
  sum_of_rolls[i] = sum(rolls) 
}

# display the results of the first few repetitions
head(sum_of_rolls)
[1] 12 14  6 12 17 11
# count repetitions where sum is 9
sum(sum_of_rolls == 9)
[1] 1230
# proportion of repetitions where sum is 10
sum(sum_of_rolls == 10) / N_reps
[1] 0.125
# tabulate the results
table(sum_of_rolls)
sum_of_rolls
   3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18 
  45  130  287  481  685  948 1230 1250 1251 1157  959  704  418  268  136   51 
# create a "spike" plot (type = "h")
plot(table(sum_of_rolls),
     type = "h",
     xlab = "Sum of three rolls of a fair six-sided die",
     ylab = "Number of sets of three rolls")

running_frequency_of_10 = cumsum(sum_of_rolls == 10)

running_relative_frequency_of_10 = running_frequency_of_10 / (1:N_reps)

plot(1:N_reps,
     running_relative_frequency_of_10,
     type = "l",
     xlab = "Repetition number",
     ylab = "Long run relative frequency of 10")

Example 1.3

As of Jun 17, FiveThirtyEight listed the following probabilities for who will win the 2022 World Series.

Team Probability
Los Angeles Dodgers 20%
New York Yankees 20%
Houston Astros 9%
San Diego Padres 8%
Other

According to FiveThirtyEight (as of June 17):

  1. Are the above percentages relative frequencies or subjective probabilities?

  2. The Dodgers are how many times more likely than the Padres to win?




  3. What must be the probability that the Dogders do not win the World Series? How many times more likely are the Dodgers to not win than to win (this ratio is the odds against the Dodgers winning).




  4. What must be the probability that a team other than the above four teams wins the World Series? That is, what value goes in the “Other” row in the table?




Example 1.4

In each of the following parts, which of the two probabilities, a or b, is larger, or are they equal? You should answer conceptually without attempting any calculations. Explain your reasoning.

  1. Consider a Cal Poly student who frequently has blurry, bloodshot eyes, generally exhibits slow reaction time, always seems to have the munchies, and disappears at 4:20 each day. Which of the following events, \(A\) or \(B\), has a higher probability? (Assume the two probabilities are not equal.)

    1. The student has a GPA above 3.0.
    2. The student has a GPA above 3.0 and smokes marijuana regularly.
  2. Randomly select a man.

    1. The probability that a randomly selected man is greater than six feet tall.
    2. The probability that a randomly selected man who plays in the NBA is greater than six feet tall.
  3. Randomly select a man.

    1. The probability that a randomly selected man who is greater than six feet tall plays in the NBA.
    2. The probability that a randomly selected man who plays in the NBA is greater than six feet tall.
  4. Flip a coin which is known to be fair 10 times.

    1. The probability that the results are, in order, HHHHHHHHHH.
    2. The probability that the results are, in order, HHTHTTTHHT.
  5. Flip a coin which is known to be fair 10 times.

    1. The probability that all 10 flips land on H.
    2. The probability that exactly 5 flips land on H.
  6. In the Powerball lottery there are roughly 300 million possible winning number combinations, all equally likely.

    1. The probability you win the next Powerball lottery if you purchase a single ticket, 4-8-15-16-42, plus the Powerball number, 23
    2. The probability you win the next Powerball lottery if you purchase a single ticket, 1-2-3-4-5, plus the Powerball number, 6.
  7. Continuing with the Powerball

    1. The probability that the numbers in the winning number are not in sequence (e.g., 4-8-15-16-42-23)
    2. The probability that the numbers in the winning number are in sequence (e.g., 1-2-3-4-5-6)
  8. Continuing with the Powerball

    1. The probability that you win the next Powerball lottery if you purchase a single ticket.
    2. The probability that someone wins the next Powerball lottery. (FYI: especially when the jackpot is large, there are hundreds of millions of tickets sold.)