MATH M3 Project - Probability Project (3)
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Apr 3, 2024
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Uploaded by Diamonddaniella0993
Simmons 1
Should I Stay or Switch?
Diamond Simmons
Colorado Community Colleges Online
MATH 1260
M3 Probability Project
March 21, 2024
Simmons 2
Before I watched the video, I did not have much knowledge of this game show, how many contestants there where, or even how you play, so the probability of wining the car to me was maybe 1/5 or 20 percent. Most of the game shows I have watched have about 5 people trying to win a prize and only 1 of those 5 people would be able to win and take the prize home with them. I grew up watching shows like Family feud, Jeopardy, and Wheel a Fortune in which all those shows have multiple players going for the same prize. After watching the video, I realized what the correct answer is to the question “What is the probability of winning the car?”. Before I believed that the probability of winning the car was
1/5 or 20% but contestants have a 2/3 or 67% chance of winning the car. After they’ve chosen the door that they believe contains the car the host, Monty opens a different door that contains a goat and proceeds to ask the contestant if they would like to pick another door. When Monty proceeds to open one of the doors that doesn’t contain the grand prize this changes the contestant’s probability from 1/3 to 2/3 chance of choosing the car. The contestants then use this to their advantage if they decide to choose a different door. If the contestant does not choose to switch, they then have the expected 1/3 chance of winning the car, since no matter whether they initially picked the correct door, the host Monty will show them a door with a goat.
Probability and probability rules can be used in many ways in a multitude of areas when it comes to seeing a person’s probability of winning or not winning. For example, when playing the card game “Pit” there are no turns, and everyone plays at once. Players trade commodities among one another by each blindly exchanging one to four cards of the same commodity. The trading process involves calling out the number of cards one wishes to trade until another player holds out an equal number of commodity cards. If there are only 8 “grain” cards and you hold 7 of them while playing with 4 people you know that this is a 1 in 4 chance that you will receive
Simmons 3
that winning card (Pit Card Game, 2024). The probability idea that this represent is conditional probability. Conditional probability is known as the possibility of an event or outcome happening, based on the existence of a previous event or outcome. The probability that two things happen together x and y is the probability of x given that y happens times the probability of y happening by itself (Barone, Adam, 2019). The formula that can be used for this problem is P (x
| y) = P(y/x) multiplied by P(x)/ P (y), the probability that the new car is behind door number
1 given that the host opened door number 2 is the probability that the host opened door number 2
given that the new car is behind door number 1. When looking at this problem X equals the new car and Y equals the host Monty opening door number 2. If the car is behind door number 1 the probability that the host opens door number 2 or door number 3 is 50/50 or 1/2. The probability of X that the car is behind door number 1 is 1/3 and the probability of y, that the host opened door number 2 or 3 is again 1/2. This problem would be worked out as follows, Finding the probability of X (that the car is behind door number 1):
P (x & y) = P (x | y) * P (y)
P (x | y) * P(y) = P (y | x) P (x)
(Divide each side by P(y) P (x | y) = P (y | x) * P (x)/ P(y)
P (x | y) = P ( ½ ) * P (1/3) / P (1/2) (The ½ cancel out) That means that there is a 1/3 chance that the car is behind your initial choice and a 2/3 chance that the car is behind door number 2 or 3 depending on what door was opened by the host (Monty Hall Problem, 2014). This is why the contestant should switch their choice since there is a higher chance of the car being behind the other door.
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Simmons 4
I was a bit surprised when I learned that the correct answer would be to switch the door from the one that they chose to another. This surprised me because I would have thought that they would recommend or talk the contestant into keeping their first option, so they did not have to give away a free car. It all made since after watching the videos provided.
Simmons 5
References Barone, Adam. “Learn about Conditional Probability.” Investopedia
, 2019, www.investopedia.com/terms/c/conditional_probability.asp.
Monty Hall Problem (extended math version). (2014, May 23). Retrieved from https://www.youtube.com/watch?
annotation_id=annotation_3286749133&feature=iv&src_vid=4Lb-
6rxZxx0&v=ugbWqWCcxrg
“Pit Card Game, Bull and Bear Edition.” Americanhistory.si.edu, americanhistory.si.edu/collections/nmah_323758. Accessed 19 Mar. 2024.
Should I stay or should I switch doors? (n.d.). Retrieved from https://ed.ted.com/featured/PWb09pny