MATH 111 Counting and Probability Project

1. Counting Techniques:

Find a situation where you would order or choose something in at least 3 stages. This could be ordering a coffee drink where you would have to specify at least 3 different options (drink type, size, type of milk …) or a sandwich (bread, meat, cheese …) It could also be something like ordering a computer and you have to choose the monitor size, memory, hard drive, etc.

a. Explain the situation you chose and list the options for each stage.

  1. Calculate how many different possible configurations there are:

i. with all possible options available

ii. if you rule out at least 1 option at each stage

iii. if you already know what 1 of the options will be (decided one of the stages)

Using the same situation you chose above, there should be the ability to add some optional items (toppings, flavors, accessories…). Separate the optional items into 2 categories if possible. For example, if you were ordering a coffee drink, you might be able to add either flavors or toppings, and if you were ordering a computer, you might have audio/visual accessories or travel accessories.

  1. Determine how many options are available and list them in each category.
  2. Decide whether or not the order the options are chosen matters to you. Explain!
  3. Calculate how many ways you could choose 3 different optional items:
  1. if they can come from any category (any 3 optional items)

ii. if you want to choose 3 items, some from 1 category and some from the other

f. Calculate how many overall configurations there can be using the stages above and
3 optional items (some from 1 category, some from another).
2. Empirical Probability:

For this part of the project, you are going to determine some empirical probabilities by collecting at least 30 pieces of data. The goal is to try to figure out the probability of some event occurring that can’t be measured theoretically and see if there are any factors that might influence the probability.

You might ask 30 different people a question, or observe 30 transactions at a store, or count up 30 objects. For each of the 30 or so pieces of data you collect, you want to measure 2 different characteristics so you can set up a 2 x 2 contingency table. This could be the person’s answer to the question and their gender or age group, or it could be 2 different characteristics of a physical object (color, type, price).

For example: I could try to come up with an estimate for the probability that a Math 111 student is full-time by asking 30 students their opinion and also recording their gender.
So the 2 characteristics I’d be using here would be:
1. theirregistration status (full-time/part-time)
2. their gender (male/female)

a. Set up a 2 x 2 contingency table to represent the data you collected.

b. Calculate the probability of each of the 4 characteristics.
(for my example, I would figure out the probabilities of each gender and each
registration status, 4 probabilities in total)

c. For each row of your contingency table, calculate the 2 conditional
probabilities relative to each of the characteristics given by the columns.
(in my example, I would calculate the probabilities that a student is full-time given
that they are female and also given that they are male, and then also for students
who are part-time. 4 conditional probabilities in total)

d. Pick one characteristic from your example to use for the following:
For example, I could pick the characteristic “part-time” or “male” or any of the 4
characteristics for my example.

i. Calculate the probability that 3 events in a row will all have the same
characteristic you chose (ie., 3 in a row that love math)

ii. Calculate the probability that at least 1 of 3 events in a row will all have the
same characteristic you chose