Investigation 2 Reflection Blogspot

1) Describe how you can find the theoretical probability of an outcome. Why is it called a theoretical probability?

First of all, let me explain what theoretical probability is and how it got its name. Now, theoretical probability and experimental probability is almost the same thing. However,there are some prominent differences between the two types of probability methods. For example, while experimental probability is when you conduct an experiment with many trails to figure out the outcomes of the experiment and collect data to help you make further decisions. Theoretical probability is found by using calculations based on data already provided, to come up with a theoretical probability of an experiment's outcomes and data. For example, lets use a fair two-sided coin and try to find out the theoretical probability of a coin toss. You already know that a  coin has 2 sides: heads and tails. The coin is also fair and is not biased towards any of these two sides. Therefore you have a 1/2 chance of the coin landing on heads and a 1/2 chance of the coin landing on tails, therefore there is a 50-50 chance for the coin to land on either heads or tails. So in the end the theoretical probability would be written like this in probability notation:
P(heads)= 1/2

P(tails)= 1/2

2. a. Suppose two people do an experiment to estimate the probability of an outcome. Will they get the same probabilities? Explain. 

This depends on the actual experiment conducted between two people as well as the number of trials that were performed in the experiment itself. It is possible that both people get the same results, or at least get results similar to each others. For example, let's say that there are two girls, one named Lizzy and the other one named Katie who decided to toss a coin 100 times.  The theoretical probability of the coin landing on either heads or tails is equal, 50-50 chance. So, when flipping the coin, you would believe that the results would show that heads and tails are exactly equal and that the coin would land on heads exactly 50 times and land on tails exactly 50 times. However, the experimental probability may show otherwise as  it's not certain that out of 100 flips, there would be exactly 50 heads and exactly 50 tails. But, it is extremely likely that the outcome will be close to 50 heads and 50 tails. To answer the question, it is not 100% certain that the two girls (Katie and Lizzy) who performed the experiment will get the exact same results. The reason I say this is because their outcomes all depend on how the experiment itself is set up (how many trials are performed).
Here is an example of an experiment that was performed to prove my explanation above. There were 100 trails of which Katie laded 36 heads and 64 tails. Of which Lizzy landed 42 heads and 58 tails. As you can see based of this information it is not always likely that the information will be spot on, in this example the information is actually quite off, but even though this is the case, it doesn't mean that the game wasn't fair. 

b. Two people analyze a situation to find the theoretical probability of an outcome. Will they get the same probabilities? Explain. 

Yes! These two individuals should get the same theoretical probability as the theoretical probability is only based on what the probability should be in THEORY not the experimental or actual probability based on the numbers of trials/outcomes.  Therefore there  is no reason which would change ones result to be different when finding the theoretical probability unless someone make an errors while calculating or just don't understand the concepts. However, apart from that, there is no reason for the results of the theoretical probability to differ. So for example, if they are examining a spinner with four colors (yellow,pink,green,purple), all colors of equal size, and we are finding the theoretical probability of getting yellow on every spin, then both people's theoretical probability should end up the same (1/4 or 25%0. The theoretical probability of the spinner landing on yellow is 1/4. This is because all four colors are equal size and the chance of landing on yellow on each spin should be 1 out of 4 or 1/4 or 25%. This theoretical probability will always stay the same. However, if we find the experimental probability based on 50 trial, the experimental probability of getting yellow will most likely be different for both people and most likely will be different than 1/4 (but very close to it).

c. One person uses an experiment to estimate the probability of an outcome. Another person analyzes the situation to find the theoretical probability of the outcome. Will they get the same probabilities? Explain.  

The experimental probability of an outcome is likely to be similar to the theoretical probability, but will not be exactly the same. If we use the same example from question 2a, the theoretical probability that Katie/Lizzy would land on heads would be 1/2 or 50/100. However, neither Katie/Lizzy actually got exactly 50/100 for heads/tails. Each of their experimental probabilities were different. The results based on Katies's set of data shows there was a 36% chance that her coin would land on heads after being tossed. 36% is not equal to the theoretical probability which is 50%. This shows that it is also likely that they will not get the same probabilities when finding the experimental and theoretical probabilities. But keep in mind that it is likely that both Katie and Lizzy will get probabilities close to one another as sometimes the experimental probability can be affected by how you conduct the experiment and how many trials you conduct. In general the greater number of trials you conduct the experimental probability will be more accurate and closer to the theoretical probability. Overall, it is quite unlikely that the people will get the exact same outcome when finding the theoretical and experimental probabilities. For example, if Katie does only two trials and gets heads both times then the experimental probability would be 100% which is far away from theoretical probability of 50%. As the number of trials increases the theoretical probability becomes more accurate.

4. What does it mean for a game to be fair?

For a game to be fair, there must be  an equal chance of both losing and winning amongst the players. This doesn't necessarily mean that each player has to have a 50% chance of winning and 50% chance of losing in order to make the game fair. It just means that each player must have an equal chance of winning, so that means that the game can't be biased. For example, if we use the same spinner we used in 2b and there are four players, and each player wins when they get one specific color on the spinner. Then the chances of each player winning are 25% and so the game is not biased against any one player. Another example is if you use a six sided dice. Player one gets a point if the dice lands on the number 4, and player two gets a point if the dice lands on the number 3. The chances of player 1 landing a 4 and player 2 landing a 3 is equally likely (based on theoretical probability). THe reason I say this is because there is a 1/6 chance that the dice will land on a 4 and a 1/6 chance that the dice will land on a 3 (theoretically). Therefore this example is equally likely or the game if fair.

5. What is a sample space and how can it be represented?

A sample space is a list of all the possible outcomes from an experiment or any other possible situation. This means that you have a list of all the possibilities written down. Generating a sample space can be very helpful when finding the likelihood of an outcome as you can spot all the possibilities. A sample space will help you get a deeper understanding of what the game could look like and show you if the game is either fair or not fair. Generating a sample space is actually very easy if you remember to draw a  tree diagram. The picture below is an example of a tree diagram. This diagram shows the possible outcomes when tossing three coins. Another way to show a sample space is by creating a matrix.

Another example would be rolling two dice. Then the sample space will look like the following matrix diagram. I got the diagram from math.hawaii.edu. Each bracket represents a possible outcome. There are 36 possible outcomes and they are represented in the matrix diagram below.