3  Interpreting Probabilities

• A probability takes a value in the sliding scale from 0 to 100%.
• Don’t just focus on computation; always remember to properly interpret probabilities.

Example 3.1

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. 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.

2. Randomly select a baby girl who was born in 1950.

   1. The probability that a randomly selected baby girl born in 1950 is alive today.
   2. The probability that a randomly selected baby girl born in 1950, who was alive at the end of 2020, is alive today.

• When interpreting probabilities, consider the conditions under which the probabilities were computed, in the proper direction

Example 3.2

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. 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.

2. 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.

3. 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.

4. 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)

5. 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.)

• When interpreting probabilities, be careful not to confuse “the particular” with “the general”.
  • “The particular:” A very specific event, surprising or not, often has low probability.
  • “The general:” While a very specific event often has low probability, if there are many like events their combined probability can be high.
• Even if an event has extremely small probability, given enough repetitions of the random phenomenon, the probability that the event occurs on at least one of the repetitions is often high.
• In general, even though the probability that something very specific happens to you today is often extremely small, the probability that something similar happens to someone some time is often quite high.
• When assessing a numerical probability, always ask “probability of what”? Does the probability represent “the particular” or “the general”? Is it the probability that the event happens in a single occurrence of the random phenomenon, or the probability that the event happens at least once in many occurrences?
• Also distinguish between assumption and observation. For example, if you assume that a coin is fair and the flips are independent, then all possible H/T sequences are equally likely. However, if you observe the coin landing on heads on 20 flips in a row, then that might cast doubt on your assumption that the coin is fair.