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Combining team probabilities

Mathematics Asked by Gav on December 6, 2021

I’m trying to establish what the combined probability would be for a certain event to occur, where there are 2 teams involved, and I know the probability of the event for EACH team. The event is ‘bookings’ within a football match. In addition, I’m also trying to establish what impact on the probability a third party would have (the referee), as they are a contributing factor to the event of a player being booked (shown a card).

The following is what I’m looking to establish:

  1. The probability of 4+ bookings the match
  2. The probability of 2+ bookings to Team A in the match
  3. The probability of 2+ bookings to Team B in the match
  4. The probability of 2+ bookings to Team A & 2+ bookings to Team B in the match

The following is the data that relates to the recent 5 matches played both Team A & Team B (not against each other), plus the most recent 5 matches the referee has refereed.

Team A received cards in previous 5 matches:

  • Match 1 (2 bookings)
  • Match 2 (1 booking)
  • Match 3 (2 bookings)
  • Match 4 (3 bookings)
  • Match 5 (2 bookings)

So, 2 or more cards shown to Team A occurs 80% of the time (0.8)

Team B received cards in previous 5 matches:

  • Match 1 (1 booking)
  • Match 2 (4 bookings)
  • Match 3 (2 bookings)
  • Match 4 (2 bookings)
  • Match 5 (0 bookings)

So, 2 or more cards shown to Team A occurs 60% of the time (0.6)

Referee cards shown in previous 5 matches (assume no team bias):

  • Match 1 (1 booking)
  • Match 2 (4 bookings)
  • Match 3 (5 bookings)
  • Match 4 (4 bookings)
  • Match 5 (6 bookings)

So, 4 or more cards are shown to either team 80% of the time (0.8)

Hopefully the above makes sense. Any help would be greatly appreciated.

Regards G

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