Question

Let’s say you were selected to participate in a TV game show. One of the games you will be playing is Cash or Nothing; where you are showed three curtains, two of them have nothing behind and just one of them has one million dollars in cash. First you have to pick a curtain, curtain #1, curtain #2, or curtain #3. Once you’ve made a selection the host will open one of the curtains that had nothing behind it and then offer you the chance to stay with the original choice or change curtains. (Note: that the host only opens a curtain that did not had the prize behind it)

a. Using Excel (or the software of your preference) do a Monte Carlo simulation for all the possible scenarios. Generate 100 replicas in the first sheet, 10,000 in the second, and 100,000 in the third. What’s the probability of winning if you chose to stay? Hint: use Excel functions and(), randbetween(), and if(). (Tip: Use one column to select the door, next column for the winning door and another column to say if you stayed or changed the selected door. Last door to say if you won)

b. Based on the results from part a, state which option is better: stay with your curtain or change. Justify the results from part a. Is this result intuitive, why or why not?

Answer 1

To determine the probability of winning if you choose to stay or change curtains in the Cash or Nothing game, you can use a Monte Carlo simulation in Excel. This involves setting up a table, generating replicas using the RAND() function, and calculating the probability of winning if you stayed by dividing the number of wins by the total replicas.

Step-by-step explanation:

The probability of winning if you choose to stay or change curtains can be determined through a Monte Carlo simulation using Excel. First, set up a table with three columns: one for your initial choice of curtain, one for the curtain with the prize, and one for whether you stayed or changed curtains.

Generate the specified number of replicas and use the RAND() function to randomly assign values for the curtain and stay/change variables. Next, use an IF() function to determine if you won or not based on your choice and whether you stayed or changed curtains. Finally, calculate the probability of winning if you stayed by dividing the number of wins when staying by the total number of replicas.

Probability is a measure of the likelihood that a particular event will occur. It is expressed as a number between 0 and 1, where 0 indicates that the event will not occur, and 1 indicates that the event will occur.

Here are some key concepts related to probability:

Probability of an Event (P): The probability of event A is denoted by P(A) and is the likelihood that event A will occur.

Sample Space (S): The sample space is the set of all possible outcomes of an experiment. Each outcome is an element of the sample space.

Event: An event is a subset of the sample space, representing a particular outcome or a combination of outcomes.

Probability Distribution: A probability distribution describes how the probability is spread over the possible values of a random variable. It can be represented as a table, graph, or mathematical formula.

Conditional Probability: The probability of an event occurring given that another event has already occurred is called conditional probability and is denoted by P(A | B), where A and B are events.

Independent Events: Two events are independent if the occurrence of one event does not affect the occurrence of the other. The probability of both independent events occurring is the product of their probabilities.

Bayes' Theorem: Bayes' Theorem is a formula that describes the probability of an event, based on prior knowledge of conditions that might be related to the event. It is often used in statistics and probability theory.