Thursday, March 25, 2010

Teaching the Concept of Conditional Probability

I still remember it like it was yesterday. Paulina was three. We picked two cards each at random from the top of a deck. The winner of the hand was the one with the highest total. We would repeat until we had gone through the entire deck. The winner of the game was the one who had won the most hands. The thing is that Paulina could not handle losing. She did not understand that this game involved zero skill. I explained it countless times, but it did not matter. Her competitive spirit got in the way. We stopped playing games of chance after a short while because Paulina could not deal with randomness. I felt at the time that we would never be able to play games of chance. Fortunately, I was wrong. She now enjoys thinking about probability and gets the idea of computing them to take good bets. Given her new-found fondness for probability, I decided to try teaching her conditional probability.

Prerequisites:
  1. understand fractions
  2. understand the concept of outcomes - The possibilities for the given problem. An outcome is something that may happen. It does not necessarily mean that it has happened.
  3. understand the concept of sets and events - Once needs to understand what sets are to tackle events. An event is just a set of outcomes. An event is used to narrow down the set of possible outcomes in a given probabilistic universe.
  4. understand how to compute probabilities in finite spaces - Know how to compute probabilities by counting. We only need the very simplest computational skills at this point.
Step 1 - Drawing Probability Trees
Teach your child how to draw spaces of probability outcomes using trees. I have found that they are simpler for young children to understand than simply listing out all possible outcomes.

Example 1: Assume that a family has two children. What are all the possible outcomes of pairs of children?

Moving from the top to the bottom, we can now read out the four outcomes: {BB, BG, GB, GG}.

Example 2: Assume that a family has three children. What are all the possible sets of children?


Once again, by reading from the top to the bottom of the tree, we can list all the outcomes {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}.

It is very important at this point to make sure that your child understands how to compute the number of outcomes without actually listing them out.

Example 3: Introduce an example of choosing randomly without replacement. This means that once an object is picked, it cannot be picked again. I would suggest using colored M&Ms here. Use two green and two red M&Ms. Explain that you want to build an outcome tree as follows. Pick one M&M at random. Leave it out of the box. Pick another M&M at random. Now, draw the tree of outcomes.

The key point when drawing this tree is to realize that once an M&M has been picked, it remains out of the set. Hence, there are only three choices when picking the second M&M. This means that the full set of outcomes is {GG, GR, GR, GG, GR, GR, RG, RG, RR, RG, RG, RR}.

Step 2 - Teaching the Difference Between Normal and Conditional Probability
Most people never really learn the concept of conditional probability because their teaches did not understand it either. Students are usually taught the formula P(A|B) = P(A and B) / P(B). The issue is that this makes little sense to most people. This is sad because math is about concepts, not symbolic manipulation. The only difference between regular probability and conditional probability is the set over which one computes.

Compute the probability of getting a green followed by a red in the last example. Looking at the tree corresponding to the last example, this event is equal to {RGG, RGG, RGG, RGG}. This is the case because there are four paths in the tree with RGG outcomes. Hence, the probability of one green followed by a red is 4/12 or 1/3.

Compute the probability of getting a green on the second pick given that the first pick was a red. This conditional probability is computed by counting outcomes in the sub-tree with Rs in the first pick. This is because we are told to assume that the first pick was an R.

This can be computed by looking at the correct portion of the tree, which is circled in red below.



The possible second picks when the first one is an R are {G, G, R, G, G, R}. Hence, the conditional probability is 4/6 or 2/3.

Step 3 - Repeat Many Times
The trick to teach conditional probability to a kid is to give lots of concrete examples.

I hope this post helps you in some way.

3 comments:

Leela said...

Dr. Perez-Fernandez-
Thank you for the blog. I really enjoy reading your posts. We just found out our daughter is highly gifted. She is 4 almost 5. Can you give me any ideas on how to teach her math. She loves numbers and keeps writing/talking about them. I am just not sure where to start.
Thank you,
Lisa

Pablo A. Perez-Fernandez said...

Leela, I did not do much with Paulina until right after she turned 5. What we did do was to play lots of games. Here are some ideas:

1. TicTacToe - This is a good game for a very young kid. Teach the kid the basic rules. Play a lot with him or her. Let the kid figure our strategy. Is it better to play first? Is is better to play in the center or the corners?

2. Checkers - This is a little more complicated than tic-tac-toe, but you will find out that the kid will get into it if he learns the basic rules.

3. Rumis - The game comes with setups of varying difficulty levels, but some of the easy ones could be good for young kids. We play this game a lot at home.

4. Blokus - We play this game a lot too. Rules are simpler, but it also requires learning how to develop a strategy.

5. Chess - I played a ton of chess when I was growing up, but the concentration and time required is not for everybody. The other games above are probably better choices for very young kids since the whole family can play them together.

I think the best thing to do before 5 is to play lots of fun, thinking games. Don't worry too much about formal education. We started when Paulina

Pablo A. Perez-Fernandez said...

We started when Paulina turned 5. She learned so quickly that I don't think she missed much by simply playing lots of games until she turns five.