# Teaching probability by using a deck of cards

I plan to teach two sessions of probability to 11th grade students using a deck of cards. My classes will be next week. I have already taught them the basic notions of writing sample spaces, computing conditional probabilities and so on.

The problem is that once students get 'trained' to use conditional probability, they use it all the time and do not go back to first principles. A typical example is the question: "What is the probability that the third card drawn from a pack of cards is a queen?". The interesting thing is that my untrained 9th grade cousin could answer immediately when I actually used a pack of cards to ask her the question. So that motivated me to teach a couple of classes on probability with a simple apparatus like a deck of cards.

Presently I have the following ideas:

Let us call the question 'What is the probability that the 4th card is a queen?' as THE question.

• Deal 5 cards face down and ask THE question.

• Now flip the third card open and ask THE question.

• Now flip the flipped card and shuffle the 5 cards and place them in some order face down and ask THE question.

• Now add two extra black suit cards (tell them the cards are from a black suit) from the deck to the array of five cards and ask THE question.

• Now tell them that I might have lied about the suit of exactly one of the cards in the previous round and then ask THE question.

Further, I plan to do the Monty Hall puzzle and Bertrand's box problem with the pack of cards. I wanted to do a basic gambler's ruin too. But I do not know how to go about it.

My question, therefore, to the community is:

1) Would you kindly suggest interesting probability questions using a deck of cards?

2) Are there interesting questions which cannot be asked using a deck of cards? If so, what simple apparatus would I need?

P.S: Buffon's needle problem would have been a good suggestion, but the students cannot appreciate continuous sample spaces as of now. That is, I would like examples from discrete sample spaces.

Thank you :)

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The version of 'Probability that 4th card is queen' that I use, is 'There are 5 chocolate chip cookies and 10 white chocolate cookies in this jar, what is the probability that the 3rd cookie that I draw out is a chocolate chip cookie?' This is not immediately obvious to be $\frac {5}{15}$. The best part is after we're done, there are cookies to eat. You could use red / green icing cookies to make it easier to see.

For the Monty Hall problem, I once ran an interesting experiment with a group of Middle School Teachers. I explained the problem to them, and there was immediate disagreement to switch or not. The next day, I asked them to position themselves in the classroom to indicate their beliefs on switching (i.e. those on the extreme left believe we should always switch, those on the extreme right believe we should never switch.)

I then ran the Monty Hall scenario, using 3 students up at the board to represent the 3 doors, gave them cards of 'Goat', 'Goat', 'Car' (pictures would be great fun). Each round, get 1 student to pick a door, I revealed a Goat, and them made him/her choose to switch or not. After several rounds, I swapped out the 3 'Door students'. This went on for half an hour.

What is interesting that you start to see a lot of movement in the classroom, as they change their minds about what to do. Their movement should be biased towards the probable outcome, so if you run it enough times, they should converge on the left side. Also, the 'Door Students' tended to have a good idea, as they are able to see who has the car/goat. While they would tend to stand on the left after retiring from their post, this information doesn't seem to get influence the decisions of the class.

Such an 'experiment' is great with well known problems that generate huge disagreement.

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