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Why I "Like" a Post Since I joined this Forum, I have been able to freely communicate with people around the world for the 1st time in my life, mostly about our common enjoyment of the wonderful

Ohhh can we start to pick on the Britts in this thread also? in the bible there is a story where it rained for 40 days. It was described as a disaster. In Enland they call it ”summer”…

Well, if that was the topic, we'd get to 100 pages in no time!

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The second video might be a clearer explanation.

Especially when it uses 100 doors to illustrate the solution.

No maths/probability calculation needed.

Shows it with a simple picture.

Hope everyone is enjoying this very famous puzzle.

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Sure but it is really that simple. picking a door has nothing to to with odds. Picking a door is arbitrary unless you have some insider info. You pick a door and then bet that you picked the wrong one because there is a 66.6% chance that the first door you picked is wrong. Simple :cheers:

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Rocky, yes, the reasoning is correct.

 

Contestant has picked one door, has 1/3 chance of winning the car.  Think of this as (Group 1)

 

Therefore, the other two doors have 2/3 chance of winning the car.  Think of this as (Group 2)

 

The host Monty Hall has just revealed that one of those doors in (Group 2) is not the winning door.

Therefore that last remaining door in (Group 2) has the 2/3 chance of winning the car. 

This is the "2/3 probability concentrated" into that door.

Change your pick and take it.

__________________________________

 

Another way of viewing the puzzle is this:

 

If you had the above choice of (Group 1) or (Group 2), you would select (Group 2), because it has two doors with 2/3 chance of winning.

When Monty Hall shows the losing door in (Group 2), and asks if you want the other door in (Group 2), you take it because it is part of (Group 2).

Edited by Paul A
typo
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https://nypost.com/2021/09/27/lisa-banes-killer-brian-boyd-charged-with-homicide/

September 27, 2021

 

As a side note:

 

Brian Boyd, accused hit and run suspect of Lisa Banes, has had charges upgraded.

Boyd was initially arrested on charges of leaving the scene of an incident without reporting and failure to yield to a pedestrian.

The new charges are second-degree manslaughter, criminally negligent homicide and failure of a driver to exercise due care.

If convicted, he could get as little as no jail time or a maximum of five to 15 years in prison. 

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10 hours ago, RockyTop said:

I started to wright down the math and it got real simple.

I got it! You keep thinking that you have three choices, You only have two. stick or change. If you stick you have a 33% chance. ......... That is it,  By swamping you get the remaining 66% They removed the other pitfall.  your remaining chance is 66%  You only really have two choices. The actual doors do not play a part in the odds. just stick or swap. 

So you have a 50/50 chance . . . which is why the 66% is so annoying - I feel like it should be 50% chance of either door - the original choice or the swap out door.

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9 hours ago, Paul A said:

Monte Carlo simulation of 100k each choice using Excel?

 

I wrote a small Javascript program for my simulation.  I've been studying JS over the lockdown we're currently enduring so this was a nice little practical exercise.  Probably not the most elegant code but I believe the logic is sound.

 

function randomDoor() {
    const rndDoor = Math.floor(Math.random() * 3) + 1;
    return rndDoor;
}
 
function findRevealDoor(prizeDoor, contestantDoor) {
    let validReveal = false;
    while (!validReveal) {
        let revealDoor = randomDoor();
        if (revealDoor !== prizeDoor && revealDoor !== contestantDoor) {
            validReveal = true;
            return revealDoor;
        }
    }
}
 
let winsorig = 0;
let lossorig = 0;
let winschan = 0;
let losschan = 0;
 
for (let i = 0; i < 100000; i++) {
    let prizeDoor = randomDoor();
    let contestantDoor = randomDoor();
    let revealDoor = findRevealDoor(prizeDoor, contestantDoor);
    if (contestantDoor === prizeDoor) {
        winsorig++;
    } else {
        lossorig++;
    }
}
 
for (let i = 0; i < 100000; i++) {
    let prizeDoor = randomDoor();
    let contestantDoor = randomDoor();
    let revealDoor = findRevealDoor(prizeDoor, contestantDoor);
    // extra code to swap contestant door
    let otherDoor = randomDoor();
    while (otherDoor === contestantDoor || otherDoor === revealDoor) {
        otherDoor = randomDoor();
    }
    contestantDoor = otherDoor;
 
    if (contestantDoor === prizeDoor) {
        winschan++;
    } else {
        losschan++;
    }
}
 
let percorig = winsorig / 1000;
console.log('Results for Sticking with Original Door');
console.log('-----------------------------------------');
console.log(`Wins: ${winsorig}`);
console.log(`Losses: ${lossorig}`);
console.log(`Percentage Wins: ${percorig}`);
console.log('         ');
let percchan = winschan / 1000;
console.log('Results for Changing to Remaining Door');
console.log('-----------------------------------------');
console.log(`Wins: ${winschan}`);
console.log(`Losses: ${losschan}`);
console.log(`Percentage Wins: ${percchan}`);
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Nice. 

Mr Elwood was also planning on running his own simulation as well.  Be interesting to see what results he gets.

Application of brute force simulations.  Effective.

The second video from time mark 2.45 shows an elegant approach that is easier, and doesn't require calculations.

________________

 

Problems can sometimes be solved without brute force.

The following can be solved without requiring pen and paper, without a calculator, without calculations.

 

A Grand Slam tennis tournament is a knockout competition.

Starting with 128 players in the singles draw.

How many matches need to be played to have an eventual winner/champion?

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1 hour ago, Paul A said:

Application of brute force simulations.  Effective.

If one breaks down the algorithm (here a bash example)

Spoiler

#!/bin/bash

cars_won_stay=0
cars_won_changed=0
tries=0

while true
do
    (( tries++ ))
    
    # Where is the car
    (( car_position = RANDOM % 3 ))

    # Pick a door
    (( door_picked = RANDOM % 3 ))

    # Stay - Choose picked door
    if [ $car_position -eq $door_picked ]
    then
        (( car_won_stay++ ))
    fi
    # Change - Choose other door
    if [ $car_position -ne $door_picked ]
    then
        (( car_won_change++ ))
    fi
    (( show = tries % 10000 ))
    if [ $show -eq 0 ]
    then
        echo "Tries: $tries Stayed: $car_won_stay Changed: $car_won_change Ratio: $(( (car_won_change *100) / car_won_stay ))"
    fi
done

one sees that

- if one stays at the choosen door, the chance to win is 1/3 - Exactly when "$door_picked" is "$car_postion"

- If one changes his picked door, after monty showed another door

Here the "winning condition" ( one picks a door, monty shows another door without the car and one changes to the third possible door) reduces to "$car_position" not equal "$door_picked".

So chances are 2/3.

Quote

The second video from time mark 2.45 shows an elegant approach that is easier, and doesn't require calculations.

So after writing the program/"designing" the algorithm, one does not need to let it run anymore...

I assume that's exactly what's explained in the video - for me personally it's easier to understand and follow if i generate the algorithm. Less possibilities to loose focus... 

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The second video shows the scenario with one hundred doors to better illustrate the solution and reasoning.

The contestant has picked one door, and has a 1/100 chance of winning the car. (Group 1)

The other 99 doors have a 99/100 chance of winning the car.  (Group 2)

Monty Hall opens up 98 doors of (Group 2) to reveal a sack of potatoes.

Monty Hall then asks if you would like to change your pick to the last unopened door in (Group 2), or would you like to stay with your original pick (Group 1).

 

This is a similar scenario to having three doors.

By using 100 doors, it seems to help people better 'see' why changing their pick is the better option.

No simulations or calculations needed, easy to visualize

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31 minutes ago, Chriull said:

Less possibilities to loose focus... 

Don’t you just hate fallacies that your mind won’t let go of, Changing negatives to positive or thinking that you have three options when you only have two. 

My calculus teacher would always preach that math is not perfect. Sometimes math screws up. I strongly disagreed. I would tell her that she is feeding the problems into the system wrong or that the system was not programmed properly. It certainly can get confusing and the best of men can’t solve every problem or spot every fallacy but math is perfect. This is the problem she gave me……….. No offense to anyone but it is very embarrassing that a calculus teacher could not see through this with someone explaining that they are switching negatives with positives. The third paragraph is an example of feeding incorrect information into the system. 
 

3 men go into a hotel. The man behind the desk says a room is $30 so each man pays $10 and goes to the room.

A while later the man behind the desk realized the room was only $25 so he sent the bellboy to the 3 guys' room with $5. On the way the bellboy couldn't figure out how to split $5 evenly between 3 men, so he gave each man a $1 and kept the other $2 for himself.

This meant that the 3 men each paid $9 for the room, which is a total of $27 add the $2 that the bellboy kept = $29. Where is the other dollar?

 

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Spoiler

The three men paid a total of: $27.

The breakdown of the $27 is:

       $25 for the room

       $2 went to the bell boy.

____________

If analyzing each individual $9

Each man paid $9.

The breakdown of each $9 is:

$8.33 for the room

$0.67 for the bellboy

________________

 

The $27 includes the $2 to the bellboy.  (Room cost $25 + Bellboy $2)

By adding $2 to the $27, it is being counted twice.

 

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Spoiler

The thing that everyone gets tripped on is that the $27 is actually a negative number. It represents money that the three men do not have. The two dollars that the bell hop has is positive. If you add them together you get negative 25, 25 is the price that the men should have paid for the room if they had not been robbed. ...... So what about the $30? ..... The three men have $3 of it. The bell hop has $2 and the hotel has $25. ..... 

@Paul A Yes, your first two statements are correct. It accounts for the money. The third, while it is not wrong holds the fallacy and the reason that people fall for the trick. Hint the third is really a restatement of the first. If you correctly add 27 and 2 together you get _____ not 29

 

Edited by RockyTop
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SINCE YOU'RE ALL INTERESTED WITH OPENING DOORS... HERE GOES A FUN RIDDLE:

A King sends you a far castle in his kingdom and tells you that if you solve a riddle before entering the castle.. It's yours with all its treasures. 

You set on you journey and after a few days you arrive in front of the castle. 

Three people are there: a wise old man and two guards. The two guards are standing, each one, in front of one door to the castle. 

The old wise man says to you:

"you have a riddle to solve so that your can enter safely in the castle and keep its treasures."

He goes on: "one if these two guards only tells lies. The other only tells the truth. And behind one door there is a bottomless pit where you'll encounter a sure and horrid death. Behind the other door there is a safe passage to the castle and its fortune."

You don't know which door is safe and which guard lies or is truthful. 

... SO, WHAT DO YOU DO TO ENTER THE CASTLE SAFELY? 

🏰🏰🏰🏰🏰🏰🏰🏰🏰🏰🏰🏰🏰

 

Edited by Paulo Mesquita
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4 minutes ago, Paulo Mesquita said:

Let's see who are the smart ones here.... 🏰🏰🏰🏰🏰🏰🏰🏰

I ask each guard in turn if the wise man is indeed wise - the one that tells me no is the liar.  

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42 minutes ago, Paulo Mesquita said:

Let's see who are the smart ones here.... 🏰🏰🏰🏰🏰🏰🏰🏰

Negative times positive equals negative. You multiply one guard by the other and get a negative answer. 

Spoiler

Ask either guard what door you want and take the other. 

 

Edited by RockyTop
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1 hour ago, Lex Smith said:

I ask each guard in turn if the wise man is indeed wise - the one that tells me no is the liar.  

Nope. 😁... But you're close on the concept. 

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1 hour ago, RockyTop said:

Negative times positive equals negative. You multiply one guard by the other and get a negative answer. 

  Hide contents

Ask either guard what door you want and take the other. 

 

Also close but not quite. BTW... This way it wouldn't work. 😁

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1 hour ago, RockyTop said:

Negative times positive equals negative. You multiply one guard by the other and get a negative answer. 

  Hide contents

Ask either guard what door you want and take the other. 

 

Theyd both give you different answers.... No good. 

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