Probability

Probability, the study of how likely something is to occur. We may not know it but probability to used everyday by all sorts of people trying to determine if an event is likely to happen. Think about your favorite toy, the company that made it used probability to determine how likely that toy is to break within a week of you starting to play with it or within a month or even a year. Probability is also used to determine how long a traffic light should be by asking how many cars are likely to build up at the light over this period of time during this portion of the day. All scientific fields use some form of probability to do their work and today we will learn a little more about what probability is.

Experiment Title:

Determining which event is more likely

Objective:

The objective of this experiment is to learn about basic probability or how likely an event is to happen through testing using six dice and seeing which is more likely 3 faces all the same or 6 faces all the same.

Background on Probability:

Pretend you have a coin and flip it. Almost always one of two things will happen: the coin will land on the "heads" side or the "tails" side. For a "fair" coin, we might say that there is an equal chance of a single throw resulting in a "heads" or a "tails". Although we cannot predict a second throw (or third, fourth, fifth, etc...), we can talk about what we might see and which events we might see more. The chances of seeing a particular event (out of all events we are interested in) can be described by a number called a probability. For a "fair" coin, we would say that there is a one in two chance of seeing a "head" and a one in two chance of seeing a "tails" given the throw of a single coin. This is the same as saing there is a 50% chance of a "heads" and a 50% chance of a "tails" if we toss the coin again.

Now pretend we have two coins, what might happen when we throw them at the same time? We can quickly write the possibilities:

-                  Both coins land on heads

-                  Both coins land on tails

-                  Coin 1 lands on heads and coin 2 lands on tails

-                  Coin 1 lands on tails and coin 1 lands on heads

We have four events, each with an equal chance of happening. With two coins, each outcome has a probability of one in four of occurring. If we were to throw two fair coins multiple times we would expect, over the long run, to see each outcome roughly an equal number of times. 

Probabilities have many more uses than coins. We might be interested in any number of questions:

-                  What is the chance it will rain today?

-                  What is the chance that a particular team will win a soccer game?

-                  If I randomly guess on a multiple choice question, what are the chances of getting it right?

-                  How many of my classmates will be absent from school tomorrow?

-                  What are the chances that a first grader, selected at random, will weigh between 50 and 55 pounds?

These questions have two similarities: All talk about uncertain events in the future and all can be defined in terms of events that may occur (and complementary events that will not occur if the event in question occurs). What are some questions that you are interested in?

Often we will not be able to come up with an exact probability of an event occurring, but may talk about a probability compared to an event that we can define a probability for. This is sometimes called subjective probability. We might even revise the questions over time based on new information. For example:

-                  Which is more likely: throwing 3 fair coins and getting three heads or that it will rain today?

-                  Which is more likely: Team A wins the soccer match or throwing a single die and rolling a six?

-                  Which is more likely: Guessing a test question correctly or throwing two coins and both landing on heads?

-                  Which is more likely: 3 or more classmates being absent tomorrow or throwing two dice and having them land on the same face?

-                  Which is more likely: a random first grader weighs between 50 and 55 pounds or a single coin thrown lands on heads?

Materials

-                  6 dice

-                  Pencil

-                  lab notebook

-                  Optional: cup to shake dice in before throwing

Method

Prepare Experiment

1. Make a prediction about which event is more likely: All 6 dice show the same number when rolled or 3 or more dice have the same number
2. Decide how many times to roll the dice. A smaller number of times may lead to more errors while a large number of throws will take more time, but provide more certainty.
3. In your lab notebook, mark an area to tally the number of rolls, tally when 3 or more dice have the same number, and tally for when all 6 dice have the same number

Run Experiment

1.      Roll he dice and tally when each event occurs

Analyze Data

Which event occurred the most? Does this make sense? Why or why not?

Conclusions

Some events occur high a higher (or lower) chance than others. This can often be shown with math or experimentally.

Make It Your Own

Which other events might be of interest? Could different combinations of events lead to different results (say throwing one or more coins and one or more dice at the same time)?

Extension Activities to do at home

Probability is useful every day for determining which events are likely to occur and which are not. In this experiment, we talked briefly about the probability of one or more independent events. Probability encompasses a lot more than independent events. To learn more, consider researching the following topics, with your parents’ permission and help in finding good sites on the subject:

-                  Joint probability for dependent and independent events

-                  Conditional probability

-                  Frequentist probability

-                  Bayesian probability