Exceptions to the Addition Rule Question

Screen Link:
https://app.dataquest.io/c/65/m/378/probability-rules/6/exceptions-to-the-addition-rule

The question on Screen 6 of 378 reads:

An online betting company offers customers the possibility of betting on a variety of games and events (football, tennis, hockey, horse races, car races, etc.). Based on historical data, the company knows the empirical probabilities of the following events:

  • Event F (a new customer’s first bet is on football) — the probability is 0.26.
  • Event T (a new customer’s first bet is on tennis) — the probability is 0.11.
  • Event “T and F” (a new customer’s first bet is on both football and tennis) — the probability is 0.03.

Find the probability that a new customer’s first bet is either on football or tennis. Assign your answer to p_f_or_t. You can’t use theoretical probability formula to solve this, so you’ll need to make use of the addition rule.

To me:

  • “a new customer’s first bet is on football” suggests that the first bet is on football and only football.
  • “a new customer’s first bet is on tennis” suggests that the first bet is on tennis and only tennis.
  • “a new customer’s first bet is on both football and tennis” suggests that the first bet is on football and tennis, and only football and tennis.
  • “Find the probability that a new customer’s first bet is either on football or tennis” suggests that we want the probability that the first bet is on football and only football, or on tennis and only tennis.

This would mean that the answer to the question is 0.26 + 0.11, not 0.26 + 0.11 - 0.03 as given in the official answer.

I understand the mathematics of the official answer – I just find the wording of the question unclear and perhaps misleading. Does anyone else agree or am I missing something?

Thank you :slight_smile:

Hi @johnny.a.hunter and welcome to the community!

You are not alone in this confusion. Your calculation (0.26 + 0.11) would be correct if your assumptions about what it means to place your first bet were correct. However, these assumptions are faulty because these two events are not mutually exclusive. Each of your assumptions are for different groups of people than what you think. Let’s use the Venn diagram below to help us visualize some of these cases.

This would be Event F. (or A in the diagram)

This is different than Event F. This would be Event F but not Event T. In the Venn diagram above, this Event is represented by A \cap B'

This would be Event T. (or B in the diagram)

This is different than Event T. This would be Event T but not F. In the Venn diagram above, this is Event is represented by A' \cap B

This assumption is correct and is represented in the Venn diagram as A \cap B

Although this isn’t represented in the Venn diagram, it is the union of two of them mentioned above: A \cap B' and A' \cap B. It would look like two Pac-Man like shapes facing each other. But this is a faulty assumption because if someone bets on football and tennis, we wouldn’t say that that person didn’t bet on football or tennis.