Figuring the Odds
Here's a simple sampling game involving a sack of labeled markers.
Player 1 randomly draws 2 markers from the sack, computes the sum of the numbers on the markers, and then replaces the markers in the sack. Player 2 repeats the procedure, and players continue alternating turns. The winner is the first to obtain seven different sums (2 through 8) at least once, or one of the sums six times.
The players keep score on a tally sheet like the ones below, which contain the results of one game played by two students in class.
As you can see, player 2 is the winner - she got all the sums at least once. What do you notice about the scores? When the students in our class looked at all the score cards from their games, they made these observations:
Most cards had more points scored in the middle columns.Some numbers hardly came up.
The x's in the columns look sort of like a bar graph.
Some players won in fewer or more moves than other players.
Some sums occur more than others.
The question is posed: What is the most likely sum to be drawn from the sack? Students formulate theories based upon the results they got in playing the game. Most students use some method requiring the adding up the number of times each sum was drawn. By using this experimental probability, it can be seen that some numbers can be predicted to occur more often than others. Based on the small two-card sample above, the following results can be seen:
The fractional values tell us the frequency of the occurences of the various sums. When we convert the fraction to a decimal or percentage, we call that the relative frequency. A histogram of the data collected from the two cards would look like this:
We can also determine the theoretical probabilities for the various sums by examining what's actually in the sack. Here's a list of the markers that are used in the game:
To determine the probabilities, we make an organized chart showing all the possible sums and the ways they can be made:
We can assign probabilities using the data in the chart, and make a histogram similar to the one we made before:
Our theoretical and experimental models compare reasonably well, although we can readily see that real-life results are not as 'tidy' as the theoretical results. This gives us an opportunity to discuss the idea of randomness as it applies to our game, and other events in real life.
We can use either model to make predictions about the game and to answer questions about it. One idea we could explore is that of deciding how many of each sum we could expect to get out of 100 (or some other number) draws from the sack. We could also think about other situations involving the game. Suppose player 1 gets a point for every odd sum, and player 2 gets a point for every even sum. Would that be a fair game? How could we make the game fair if it isn't already? By exploring this and other games involving random chance, students can begin to get a feel for using probabilities, both theoretical and experimental, to make decisions.
Assigning probabilities, or 'odds', to various events is a fundamental part of decision-making in our society. 'Risk management' is the cornerstone of corporate and institutional decision-making, whether we like it or not, and probabilities form the foundation of the process. It is important that students develop an intuitive, critical approach to assigning probabilities to events, or outcomes, and that they understand the process when reference is made to it in advertising, business and government.
