A tweet from Megan Hayes-Golding led to some fun:
The fun was that the very next day my sister had a first guess win playing Clue with the kids:
On our way out to eat last night the kids were wondering what the probability of a first guess win was!
They told me that she had three cards in her hand, but didn’t know what they were. Today we worked out the probability – turns out to be roughly 1 in 200.
This is a great problem for kids and a really nice way to reinforce some basic counting ideas.
Here’s a sketch of how we worked it out:
(1) Step 1. How many different three card hands are there? In Clue there are 6 suspect cards, 6 weapon cards, and 9 room cards, for a total of 21 cards. All of the cards are distinct, so we computed that there were 1,330 different 3 card hands ( 21 choose 3).
(2) Step 2. Compute the different kinds of three card hands, and how many of each type of hand there is. For example, we found that there were 324 3-card hands with 1 suspect, 1 weapon, and 1 room card.
After we found all of the different types of hands we double checked that this way of counting the hands also led to 1,330 different 3 card hands. It was nice to have a way to check that we’d not made any mistakes (yet!).
(3) Next we found the probability of a first guess win with each type of hand. For example, the probability of guessing right on your first guess if you have 1 suspect card, 1 weapon card, and 1 room chard is (1/5)*(1/5)*(1/8).
In case you aren’t familiar with the game, you are trying to determine who committed a crime, in what room, and with what weapon. The cards revealing this information are in an envelope at the center of the board. Each card you hold tells you a particular person / item / room that was not involved in the crime. So, if you hold exactly 1 suspect card at the beginning of the game, you have 5 suspects remaining who may have committed the crime.
(4) Finally, we found the average probability from all 1,330 cases and found that it was:
3,047,827 / 603,288,000 which is about 1 in 200 🙂
You can see on the paper that we made one arithmetic mistake computing this average. We accidentally entered 81/10 instead of 81/50 in the calculation. That error led to us computing a probability of about 1 in 100, but that answer didn’t make sense because all of the individual probabilities were less than 1 in 100, so the 1/100 couldn’t have been the average.
So, a fun project and an amazing coincidence with the earlier tweet. I think this would be a great activity for a stats class. It is also a nice basic counting example, too.