Practice Problems (Final Exam) – ISA 225

Answer: 1/3 or 0.33

To determine the probability of Event A occurring (a.k.a. "P(A)"), we must first understand that the sample space for all outcomes of a six-sided die roll look like so:

S = {1, 2, 3, 4, 5, 6}

In other words, the outcome of rolling a six-sided die can be 1, 2, 3, 4, 5, or 6.

Now, the sample space for Event A...

• Event A: Roll a number greater than 4
• Event B: Roll an odd number

...looks like so...

S_A = {5, 6}

...because we're only looking for rolls greater than 4.

From here, we can solve for P(A) with the following formula...

...which the "# of outcomes in A" equals 2...

S_A = {5, 6}

...and the "# of all potential outcomes" equals 6...

S = {1, 2, 3, 4, 5, 6}

...resulting in a P(A) value of 1/3, or 0.33!

This serves as our final answer!

Answer: 1/3 or 0.33

The marginal probabilities are the ones on the outskirts...

...as they represent the probability that one event occurs... a.k.a. row / column totals. (Because each row / column represents one given event!)

The rows represent Event #1 (whether or not the student wore a hat)...

...and the columns represent Event #2 (whether or not the student received a bid).

Answer:

To start, we can set up a blank probability tree like so...

...and assign whether or not they wore a hat to the first "round" of legs (since that's what the problem indicates)...

Question #1: Construct a probability tree (Setup: Hat ➡ Bid).

...and then assign whether or not they received a bid to the second "round" of legs (since that's what the problem indicates)...

Question #1: Construct a probability tree (Setup: Hat ➡ Bid).

From here, the prompt tells us that 75% of students who wore a hat received a bid...

Consider the following rush stats for Sigma Apple Pi: 75% of students who wore a hat during rush received a bid to Sigma Apple Pi, whereas only 20% of students who didn't wear a hat received a bid. According to brothers, 40% of students wore a hat during rush.

...meaning that the other 25% did not receive a bid.

We can fill in those values like so:

Then, the prompt tells us that out of the students who didn't wear a hat, 20% received a bid...

Consider the following rush stats for Sigma Apple Pi: 75% of students who wore a hat during rush received a bid to Sigma Apple Pi, whereas only 20% of students who didn't wear a hat received a bid. According to brothers, 40% of students wore a hat during rush.

...meaning that the other 80% did receive a bid.

We can fill in those values like so:

Lastly, the prompt tells us that 40% of students wore a hat during rush...

Consider the following rush stats for Sigma Apple Pi: 75% of students who wore a hat during rush received a bid to Sigma Apple Pi, whereas only 20% of students who didn't wear a hat received a bid. According to brothers, 40% of students wore a hat during rush.

...meaning the other 60% didn't wear a hat.

We can fill in those values like so...

...resulting in our finalized probability tree!

Answer: 0.10

In market basket analysis terms, "support" essentially means "out of all transactions, which contain both a Hoodie & a Hat?".

We can set up the following formula to determine this:

Based on the prompt, we can see that there's 1,000 transactions with both a Hoodie and a Hat...

Consider the following: Crammer Nation University Merch Store has 10,000 transactions. Out of all transactions, 5,000 of them include a hoodie. Of these hoodie transactions, 1,000 of them include a hat.

...which we can plug in like so:

Then, the prompt tells us that there were 10,000 total transactions...

Consider the following: Crammer Nation University Merch Store has 10,000 transactions. Out of all transactions, 5,000 of them include a hoodie. Of these hoodie transactions, 1,000 of them include a hat.

...which we can plug in like so:

When we solve from there...

...we get a support value of 0.10! This serves as our final answer!

Answer: 0.10