Probability (11
Probability Unit (Textbook pp. 602 – 663)
Probability theory originated with games of chance – gamblers in 17th century France wanted to improve their likelihood of winning and applied mathematical thinking to the problem. If you are interested in learning more about the history of probability theory, this LINK leads to a website that summarizes some of the people and major developments in the history of this branch of mathematics.
In general - probability tells us the relative frequency of an event or the likelihood of an event. We need to develop some technical vocabulary to help us in the discussion of this topic.
Experiment: An activity whose results can be observed and recorded.
Examples of probability experiments might be rolling a die, tossing a coin, or selecting a marble from a jar of red and blue marbles.
Outcome: One of the possible results from an experiment.
Examples of outcomes might be the number that ends up on top of a die, which side of a coin is on top, or the color of the marble you chose.
Sample space: This is the set of ALL possible outcomes for a given experiment.
Examples of samples spaces would be {1,2,3,4,5,6} for rolling a die, {H, T} for flipping a coin, or {RRRBBBBBBB} for selecting a red or blue marble from a jar.
Event: A subset of the sample space.
Examples of an event might be getting a 6 on the die, getting a heads on a coin, or drawing a red marble from the jar.
Let’s look at another example of how to use the vocabulary.
A bag contains 4 blue, 5 red, and 1 white marble.
Experiment – draw 1 marble from the bag
Outcome – look at the marble, what color is it?
Sample space: {BBBBRRRRRW }
Event – {B}
Now let’s define Probability.
The probability of an event E happening can be defined as a part-whole ratio (fraction).
[pic], where P(E) = probability that event E will occur, n(E) = number of outcomes for event E or the number of ways event E can happen, and n(S) = number of outcomes in the sample space S (all possible outcomes).
So in our examples with the red, white, and blue marbles in the bag, let’s find some probabilities using our definition.
P(draw a white marble) or P(W) = [pic]
P(draw a red marble) or P(R) = [pic]
P(draw a blue marble) or P(B) = [pic]
Note: Since probability is defined as a part-whole fraction, it can also be represented as a decimal or percent!
Now what is P(draw a purple marble)? Well there are no purple marbles in the bag unless someone is cheating. So P(draw a purple marble) = [pic]
The smallest probability you can have is 0 – which means the event is impossible.
What is P(draw a red, or a white, or a blue marble from the bag)? Use some common sense.
P(draw a red, or a white, or a blue marble from the bag) =
[pic]
We sometimes call this a sure thing!
Some basic ideas from probability theory.
P(E) = 1 means an outcome is certain – a sure thing.
P(E) = 0 means an event is impossible.
All probabilities are numbers between and including 0 to 1.
If A is an event and [pic] is it complement, then [pic].
Two types of Probability
There are two ways to think about probability.
• Theoretical probability is what the formula predicts will happen.
• Experimental (empirical) probability is a record of actually doing the experiment many times and recording the answer.
Think about it this way: We all know that the probability of getting a head when we flip a coin is ½ or half of the time we expect to get a head – that is the theoretical probability. NOW take a coin out of your pocket or purse and flip it ten times and write down what you get each time. How many heads did you actually get? Some of you probably got 5 – like we would expect. But some of you maybe got 4 or 7. When we actually DO the experiment, we don’t always get the expected theoretical probability. So what good is it? Well the theoretical probability is actually what we get when we do a certain experiment many, many times.
This is called the “Law of Large Numbers” (Bernoulli’s Theorem) – If an experiment is repeated a large number of times, the experimental or empirical probability of a particular outcome will approach the theoretical probability as the number of repetitions increases.
Going back to our coin tossing experiment. If we each tossed a coin 1000 times, our experimental probability would get very close to ½. You can simulate this with a class of students by having each child do the experiment a certain number of times – say 20 times, then compile the data from all of the students in the class – thus getting a lot of repetitions of the experiment. OR you can use computer or calculator programs to simulate the experiment. Most graphing calculators will have little programs that simulate rolling a dice or flipping a coin a given number of times.
Here is a link to a quick coin toss simulator: Simulated Experimental Coin-Toss Data
Try it out!
Let’s look at one more example. What if our experiment was to roll 2 dice, one red and one green, and add the dots on the top of each die to get a sum. What would our sample space be? Take a minute and write down what you think the sample space for the experiment would be.
[pic] OK time’s up. If you wrote {2,3,4,5,6,7,8,9,10,11,12} you are on the right track, but you don’t have the whole picture. Here is a link to a great program that simulates throwing two dice a lot of times and gives you a graph of how often different sums occur. Try it for 100 tosses, then 500, 1000, and 5,000. Do you start to get the idea that some sums are more likely than others?
>>>>>>>Link to Kid’s Zone: Chances
Let’s take the 3 in our list above. Is there only one way to get the 3? Well actually there are two ways to get the 3. We could have one dot on the red die and two dots on the green die OR we could have one dot on the green die and two dots on the red. So there are 2 ways to get a 3!
What about the 7? There are several ways to get a 7 as the sum on two dice. Can you list them all?
When you have your list, go to the next page and see if you got all of them.
A nice way to look at the sample space for this experiment is to make a little addition chart.
RED DIE
|GREEN DIE |+ |1 |2 |3 |4 |5 |6 |
| |1 |2 |3 |4 |5 |6 |7 |
| |2 |3 |4 |5 |6 |7 |8 |
| |3 |4 |5 |6 |7 |8 |9 |
| |4 |5 |6 |7 |8 |9 |10 |
| |5 |6 |7 |8 |9 |10 |11 |
| |6 |7 |8 |9 |10 |11 |12 |
The bold black numbers in the chart are the sums that make up our sample space. You can see there are more than 11 possible ways to throw two die. There are actually 6 x 6 or 36 ways to do it. Notice there are six ways to get a total of 7. It has the highest theoretical probability of occurring:
[pic]. If you played around with the two dice simulator I provided above, you will notice that the theoretical probability of getting certain sums doesn’t always match up with the experimental probabilities we get when we actually do the experiment. But the more rolls you asked the program to do, the closer the results get to the theoretical probabilities. THAT”S THE LAW OF LARGE NUMBERS!
In our next section we will develop additional methods of determining the size of the sample space for more complicated experiments.
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