3 Discoveries in Mathematics That Will Change How You See The World

My dad used to always say there's 3 types of people in this world.. Those that can count and those that can't.... Very important life lesson I think 🤔

The keys to understanding The Monty Hall Problem are realizing that the host knows which curtain hides the winner, and that when he opens a curtain, he'll always choose to reveal one of the losers (otherwise it'd ruin the game, right?) When you choose one of the three curtains at the beginning, you are also deciding which two curtains the host can choose from when it's his turn to open one. If you pick the winner initally (a 1 in 3 chance), he'll have two losers to pick from, and if you pick one of the losers (a 2 in 3 chance), he'll have one loser and one winner to pick from (being forced to pick the loser as stated above). So when he opens one of the remaining curtains and reveals the loser, there's a 2 in 3 chance he was forced to do so (because the other curtain that you didn't chose initially is the winner), meaning 2 in 3 times you chose a loser at the beginning. So yeah, you should switch. EDIT: I love the pedants insisting on applying real game show logic to a math problem: I get it, Monty could have not followed the rules. I can only imagine you in grade school: "ACKSHUALLY MRS SMITH, if Ralphie had 5 Pennies and gave 2 to Suzie, he would have 4 left because he likely palmed one!"

I actually kind of like the way Mythbusters explained the Monty Hall problem. If you consider your door is a 1/3 chance and the other two doors are a combined 2/3 chance, the fact that one of those doors is opened doesn't change the fact that they are a combined 2/3 chance.

This video hit close to home. Being color-blind in the green-red spectrum, I can't count the number of times that I have attempted to explain what the world looks like to me. People are often surprised to find that to me the world looks perfectly normal for it is the only way that I have ever seen it. I was told that certain color was green and therefore I call it green. The odds are my green differs from your green. The best way to explain how my vision varies is to show its' impact. For instance, showing up for a dress-green uniform inspection in the military wearing blue suit pants (yes, I really did that.)

The Monty Hall problem is that your choice of curtain is random - the hosts choice to show you is NOT random. The host never shows you your curtain OR the curtain with the $1,000,000. It's the combination of non-random and random variables that causes most of the confusion.

In the Monty-Hall problem, the host does not choose randomly. ... and that's it. That's the one thing that changes everything.

Another way to look at the Monty Hall problem is that it's effectively the same as giving you the option of sticking with your original choice or changing to selecting both other doors.

I think the funniest example of a Nash Equilibrium is when two people approach each other while walking around a corner. Both frighten each other and for whatever reason both step to the left and both step to the right at the same exact time. This is actually the opposite of the equilibrium. The correct method would be to converse with the other participant and say "Go to the right" or just grab them by the shoulders and do a 180 degree spin with them so neither can get confused. My method is that whenever this happens to me my first and only instinct, instead of getting stuck in the dumb left right left right loop of laughter and embarrassment, is to stand directly upright and perfectly still like a deer in the headlights. It freaks the other person out which makes them go around me faster, but also leaves no room for guesswork and they know that my intention is to let them go around me in whichever direction they want.

Great explanation of the Monty Hall problem. I think it's so confusing because it isn't a pure maths problem. The host doesn't pick a door at random, but people intuitively assume it's random chance.

The Monty Hall problem was not clearly explained in the first go. It's absolutely critical to say the host then reveals a door that ISN'T the winner based on his knowledge of which door has the winning prize. If he just shows door 2 when you choose 1 and door 3 when you choose 2 and door 1 when you choose 3 (or any other just guess) regardless of what has the prize or not then your odds don't change. But he has a chance of just revealing the winner that way and then you don't get a choice to swap anyhow. It's using that extra information that changes the odds. He must know what the winning door is.

The first lesson I learned in grad school probability was: If at all possible enumerate all the outcomes. Here, that's easy. Assuming: 1) Monty will not expose the $million prize 2) the contestants choice of prizes is equally likely 3) if Monty has a choice of prizes to expose, his choices are equally likely Without loss of generality we can concentrate on the prizes picked and shown rather than the doors -- we can always renumber the doors here. Here are the 4 outcomes: (a) Contestant picks $1,000,000 Monty shows $1,000, probability 1/6 (b) Contestant picks $1,000,000 Monty shows Banana, probability 1/6 (c) Contestant picks $1,000 Monty shows Banana, probability ⅓ (d) Contestant picks Banana Monty shows $1,000, probability ⅓ compute the expected value of never switching: (a) 1,000,000 * 1/6 keep $1,000,000 (b) 1,000,000 * 1/6 keep $1,000,000 (c) 1,000 * ⅓ keep $1000 (d) 0 * ⅓ keep banana expected value $333,667 (sum of: prizes * probabilities) Compute the expected value of always switching: (a) 0 * 1/6 switch to banana (b) 1000 * 1/6 switch to $1,000 (c) 1,000,000 * ⅓ switch to $1,000,000 (d) 1,000,000 * ⅓ switch to $1,000,000 expected value: $666,833 (sum of: prizes * probabilities) Add to the Mathematics "Strange But True" list.

a crucial piece of information was left out of the 3-door problem. It is only true IF THE HOST OF THE GAME SHOW KNOWS BEHIND WHICH DOOR THE TOP PRIZE IS (as other commenters have mentioned). If the host doesn't know (and if the host may inadvertently open a door which conceals the top prize) then the chances/probability don't change. This is easier to appreciate in the 100 door example. If the host doesn't know where the top prize is it would be a 1 to 100 chance that he or she would open all the doors except the one where the prize lies.- i.e. the same chance as the game show player has that their choice is correct.

"Math is perfect, real life is not" a very beautiful quote the represents the reality. Here is an example: I studied electronics and since I love playing bass, I decided to make an amplifier from scratch. I even got specialized equipment like an oscilloscope to get the best idea of what's going on. Here is what I found out: the equations that are supposed to give you the results of your circuit are simply not correct. Of course some DO give you the results you should expect after doing the math but things were acting very weird to the point I gave up because they made no sense and I didn't know how to proceed to complete it. Multisim, a program that allows digital simulation of a circuit would give me errors all the time because it, just like me, just couldn't comprehend wtf is going on there. The circuit outputted about 3% of what I wanted and since it wasn't any major project or something, I simply gave up and left it for when I have the will to complete it, at some point in the future, hopefully. Before you blame me, I am confident that the circuit was correct. Only faulty parts could have been the reason for this but I just bought them and they were brand new, so, I don't know.

Probably one of my favorite SP videos so far, great Job from Simon and Team. I'd love to see more content on rather complicated topics like this.

The Monty Hall problem only rewards changing your guess if it's assumed that the host knows where the million dollars is and will always open another curtain. If that condition isn't met, then the remaing two curtains each have s 50:50 chance of hiding the million dollars (and in one game out of every three, the million dollars will be behind the curtain the host opens, so you're choosing between the thiusand dollars and the banana, and you have a 50:50 chance of that)

Savvy and slightly sarcastic gamblers have a saying, "Don't bet on anything that talks". Well, that's that's the problem with the Monty Hall Problem. Rather than strictly being a math or statistics problem, it confuses things by including Monty, without explaining anything about his consistency or motives.

What, Simon actually watched a movie? We need to start making a list of the movies he's actually admitted to watching

Monty Hall you left out dome rules : Host will ALWAYS show a second curtain weather or not you have the correct one : Host will never revile the big prize.

I think that the main thing that people who explain the Monte Hall problem never really pin down into simple words is that Monte Hall would NEVER open the door to the million dollar prize on purpose. That is critical to the problem and I've never seen anyone explain the problem and mention that. Anyone who has seen the show automatically knows this critical detail.

0:45 - Chapter 1 - The monty hall problem 4:30 - Chapter 2 - The nash equilibrium 11:30 - Chapter 3 - Math might not be real