Week 2: Calculating probabilities#
STATS 60 Spring 2025
10 bonus points for attendance
Discussion assignment (extra credit, bonus points: 10 + up to 5):#
Think of a coincidence that you have observed in real life. In the following Google Form:
Describe a probabilistic experiment that models the coincidence. Be sure to explain why your experiment models the coincidence, and to justify any assumptions you made.
Calculate the probability of the coincidence that you observed. You can use a computer to do this; just be sure to explain your logic.
Are you still surprised by the coincidence?
Discussion Agenda#
Show and tell (10 minutes)#
The discussion instructor selects one example of a ``coincidence experiment’’ from their section.
For each selection, the student curator explains their coincidence.
The class discusses the example and the modeling assumptions. Keep in mind:
Does the experiment model the coincidence well?
Does this remind you of any of the coincidences we studied in class?
Monty Hall (5 minutes)#
The T.A. explains the rules of the Monty Hall game:
There are three doors, two of them hide goats, one hides a prize.
The player picks a door.
The game show host opens one of the other doors to reveal a goat.
The player has to decide if they stay or switch.
Queston: what is the probability that the player wins the prize if they stay?
Class poll: what is the optimal strategy, stay or switch? Or, does it not matter?
Hypothesis testing (20 minutes)#
Students will hypothesis test, by repeated experience, whether it is a better strategy to switch.
The null hypothesis is that switching is no better than staying.
The test:#
Students pair up in groups of 2.
Each pair of students gets 3 cards (2 black, 1 red) from a deck of cards. The black cards are “goats” and the red card is the “prize.”
Repeat for 12 games:
The “game show host” will shuffle and fan out the cards, so that the “player” cannot see the faces.
The “player” picks a card
The game show host reveals one of the remaining black cards to be a “goat”
The player switches
Record whether or not the player got the prize.
Data analysis (5 minutes):#
The class comes together, and count the number of total wins (and the number of total games).
TA walks class through a calculation: if switching is no better than staying, then what is the probability that the fraction of games won is \(\ge \frac{7}{12}\)?
Class decides whether to reject the null hypothesis.
Mathematical analysis (10 minutes):#
TA and class work through the mathematical analysis of Monty Hall.