Single-Stage vs. Multi-Stage Experiments

There are two different type of experiments that we can find the probability for, single-stage and multi-stage experiments. Single-stage experiments are exactly what they sound like, an experiment with only one "stage" or step. There is only one sample space and one probability to calculate, an example of this would be finding the probability of rolling a regular die one time and getting a two. All you have to do is find the probability for that singular roll. 

Multi-stage experiments are experiments with multiple "stages" or steps. There are multiple sample spaces and lots of outcomes to keep track of, that is why we use tools like probability trees to help us keep track of all the possible outcomes. An example of this would be rolling a regular die three times while also spinning a spinner and finding the probability of rolling a two and landing on the green space on your spinner. Multi-stage experiments are much more complicated and time-consuming than single-stage experiments are. 

Probability Trees

The picture above shows an example of a probability tree. Probability trees are a math tool used to help solve multi-stage experiments and are described as "represent[ing] a series of events that happen sequentially, with all possibilities and probabilities shown." In other words, they help you keep track of your sample space as it is much larger than it is in the single-stage experiments. Probability trees are mostly used to ensure that every possible outcome is accounted for through the different branches, with each branch being the next step in the experiment. 

Before this I had no recollection of learning how to use a probability tree, I had remembered creating something similar with food chains in high school biology, but nothing in math. However, with some more practice throughout the notes I became confident in my ability to properly use/set up a probability tree on my own. 

My Understanding of Probability

My understanding of probability has grown tremendously, I know that I learned some of the basics of probability at some point in my academic career, but not nearly as in depth as we went in this class (or at least not that I can remember). I went from feeling like I understood the concept of probability within the single-stage experiments, to feeling totally lost and confused on some of the multi-stage experiments. I specifically struggled with determing how to set up the probability tree on some of the multi-stage experiments. The wording would often trip me up and I would set up the tree the wrong way, however I have done some extra practice since then and I feel much more confident now. I now feel like I can tackle most probability problems without too much trouble.

This artifact is from notes we took on Thursday of the Week One. We had already gone over the basics of probability on Tuesday in our 8.1 notes - Single-Stage Experiments (Introduction to Probability) of Week One.

What is Probability?

Probability is described as, "the likelihood of an event occurring," and is denoted as a "P(E)." Probabilities can be expressed through fractions, percentages, or decimals, this is because probabilities are also always a value between 0 and 1, for example, 1/4, 25%, or .25. 

When talking about probabilities we use events and outcomes. An event is described as, "Any subset of outcomes of the sample space," or in other words the situation we are finding the probability for, like flipping a coin a certain number of times, and is denoted as "E." An outcome is described as, "When an experiment is performed, the result is an outcome," an example would be flipping a coin four times and getting heads once, the outcome was getting heads 1/4 times. 

Another component used when calculating probability is the sample space. The sample space is described as, "A set of data listing all the possible outcomes of an experiment." An example would be if you have three red marbles and two green ones, your sample space would be, {R1, R2, R3, G1, G2}, where R1, R2, and R3 represent the three red marbles and G1 and G2 represent the two green marbles. Sample spaces are used to help us keep track of all our possible outcomes within an event. 

Experimental vs. Theoretical Probability

There are two types of probability, experimental and theoretical, each are used for different situations. Experimental probability is described as, "uses a statistical experiment to determine the probability of a defined event," and is found by taking the "Frequency of E" and dividing it by the "Total Frequency." This type of probability is used when an actual experiment is being conducted, the outcomes do not all have to be equally likely. Theoretical Probability is described as, "based on theoretical considerations. Theoretical probabilities describe the outcomes of ideal experiments, with equally likely outcomes," and is found by putting 1 over n, the amount of equally likely outcomes, or 1/n. The main difference between the two types of probabilities is the use of equally likely outcomes, theoretical requires them, whereas experimental does not. For most of the work we did in this unit we used theoretical probability because we were not actively doing the experiments mentioned in the problems.