Probability

---

Please e-mail your suggestions...Link to my other web site.

Probability, in its most basic concept, is a measure of how many times a certain event will occur out of a number of possible events. It can become quite complicated; imagine the complexity of an actuarial's computation of the probability of having an accident for any given age group, out of the total number of times people get in an automobile and drive!

We can construct a simple model, however, which can be helpful for students in beginning to assign probabilities for events. One such model is a rectangle area model, with parts of the rectangle representing the occuring events.The fraction of the area represented by each part is the probability that a given event might occur.

The model might look like this:

Here's another rectangle:

Suppose we have a bag of colored tiles. We could draw one tile at a time and record its color, then put the tile back, shake the bag, and draw another tile and record its color. If we did this 20 times, we might end up with the following data:

• red = 6
• blue = 4
• green = 7
• yellow = 3

We could ask questions like What is the probability of drawing...

• a red tile?
• a blue tile?
• a tile that is not yellow?
• a tile that is either blue, green or yellow?
• an orange tile?

The situation can be modeled with a rectangle; to make thinking about the colors easier, we can group the colored squares together...

These results would be called experimental probability because they came from an experiment, the drawing of tiles from a bag. The probabilities would be written like this:

If we opened the bag and discovered the actual contents, we could construct a model for this, as well. Perhaps the actual contents were 5 red tiles, 5 blue tiles, 6 green tiles and 4 yellow tiles. Our model then becomes...

Now the probabilities are called theoretical probability because they can be computed from actual knowledge of what's in the sack. The theoretical probability for drawing each color would be...

The experimental and theoretical probabilities do not match because of a property called randomness. The drawing of the tiles is a random occurence. We don't know what color will be drawn on any one draw. We could have drawn tiles 20 times and never drawn a yellow tile.The theoretical probabilities can, however, tell us what to expect in our drawing process, and we can use them to make predictions.

• Suppose we were to draw tiles 100 times from sack. How many of each color would you expect to draw based on our theoretical probability? How many would you expect based on the experimental probability?

In real-life situations, we might never know what's actually 'in the sack'. We have to perform experiments or collect data in order to make predictions about certain events, like automobile crashes, failure of airplane parts, volcano eruptions, earthquakes, or life spans we can expect to live.