to ask if you can answer a question re probability (Maths question)

[Fainne]

So, say I have 20 cards in a pack.

I pick one. It's the Ace of Diamonds let's say for argument's sake.

I then pick another one out of the same pack of 20 cards the following day.

Am I correct in saying that the odds of me picking the same card is a multiple of the single odds?

So 1/20 x 1/20 = 1/400

?

Because I've someone telling me the odds are still 1/20 that I'll pull the same card.

But on the 3 door game.
You had a 1 in 3 chance of winning a prize.
Remove one door and your odds change.
You now have a new game with a 1 in 2 chance.

That only works if a door is removed at random. It isn't, a non-winning door is removed. That's the key to the problem. Monty introduces bias by specifically removing a door that doesn't hide a prize, increasing the chances that the prize is behind the door that is remaining.

I love the birthday one.

It's hard to explain to people - because it seems so strange.

I did hear the lottery justify the inclusion of more numbers as that means there were more numbers to choose from and that's a good thing.

I see the OP was talking about playing Poker.
There is quite a high chance of a pair of cards in Poker.
But a pair of Aces?

Yes I was thinking along the same lines.
That makes sense.
Picking the right door is different to winning the prize.

So would you stick with your original choice - given that there are 2 doors left?

Or would you go for the other door? The one that you didn't pick.

Or do you think it doesn't make any difference as the odds are 50 / 50?

Odds on the first day picking a random card is 20/20 (if you dont want a specific card)
Odds on consecutive days are 1/20 as you have a 1 in 20 chance of selecting the same card.
Your combined probability of pulling a predetermined card on both draws is (1/20)×(1/20)

Or if you are pulling a random card and want to then subsequently pull the same card its (1/1 or 20/20) x (1/20)

I would stick with my door,
But if I was given multiple chances to replay the game of course you swap.

One of the biggest fights my partner and I had was when I was trying to explain the Monty Hall problem to him and he would. not. accept. it. There was shouting, I drew diagrams, it was a mess xD

I eventually convinced him by demonstrating over and over with a coin hidden behind one of three tea towels.

I would stick with my door

You have a 1 in 3 chance of being right.
A 2/3 chance of being wrong.

So statistically it's better to swap.

Like if I had a biased coin. If you knew that it was heads 2/3 of the time and tails 1/3 of the time, you should pick heads.

Coverting the chance to percentage you have a 0.05% chance of pulling the same card

This has been such an enlightening thread. I've learned more about probability in these 15 pages than I did through all of senior school! Thank you to everyone who explained things so clearly. I wish I'd had maths teachers like you in school.

Could anyone recommend online resources so that I could learn more? DD has just begun with probability in school and I'd love to not panic when she shows me her homework.

One of the biggest fights my partner and I had was when I was trying to explain the Monty Hall problem to him and he would. not. accept

I hear you.

But if I was given multiple chances to replay the game of course you swap

That makes no mathematical sense.

The odds of being right are the same no matter how many times you play.

Apparently with the monty hall problem pigeons learn/realise more quickly than humans that switching their choice is the correct strategy

As many people have said above, it depends on the formulation.
(1) If you are interested what is the probability of pulling out an ace of diamonds on both days it is 1/400.
(2) If you are interested in what is the probability of pulling out the ace of diamonds today given that you pulled it yesterday it is 1/20.
Confusing (1) and (2) is very common and even has it's own name - known as "gambler's paradox".

If your doing it one time only.
You have only a 16% more chance due to probably .
The more times you do the pick the more the 16% chance matters.
The chances the second door rises each game.
2/3 first game.
4/6 second game
6/9 third game.

2/3 first game
4/6 second game
6/9 third game

Those are exactly the same fractions.

Seriously.

The odds don't change.
It will always be 2/3 to 1/3.

It about the out amount of doors opened at the odds.
The more games you play the more doors to open, the more the odds show the swap kicks in.

The more games you play the more doors to open, the more the odds show the swap kicks in

No, That's not how probability works.
You will always have a 1 in 3 chance of being right.
A 2 in 3 chance of being wrong.

The best strategy is to swap. Always.

If you lose the first time, that makes no odds.
It's a new game. Irrespective of the last one.

If you do it enough, you will find that you have won 1/3 of the time and lost 2/3 of the time. Assuming you don't swap.

If you swap every time, you will win 2/3 of the time.

Ok so say this is true.
Over a week 5 people play .
No one swaps.
All 5 win.
Does that mean the odds were wrong or just all 5 people bucked the odd?

Does that mean the odds were wrong or just all 5 people bucked the odd

It means that you haven't got enough data to get the actual probability.

If you toss a coin, you could get 5 heads in a row. It happens.

But if you do it enough times, you will get to 50%.

The trick is to do it enough times.

But unlikely things do happen.

Agreed.
That was why I said first show I stick with my box.
100 shows i'd take theirs every time.

That was why I said first show I stick with my box

Fine - but you do agree that you have less chance of winning the prize than if you swapped?

I think that you are telling me that if you knew a coin was biased towards heads and you were asked to choose, you would choose tails.

Even though you know it is more likely to be heads.

Fancy a game of poker?

Yes and no.
The game was 3 doors.
That game had a 1/3 chance of me winning. While there is now more probably of their door having the prize, the game is now out of two door and that means I am now also being give a 50/50 chance of getting the right door.