Monty Hall
You enter a game show. When it starts you see three doors.
You are promised that behind one of the doors is a car, and behind the other two doors are goats, and you will go home with the item behind the door you choose.
The host, Monty, asks you to choose a door. You pick one of them, call it ‘A’.
Then Monty opens one of the other doors and shows you a goat behind it.
Then Monty asks if you would like to change your chosen door at this point to second other door, or keep your original choice. He promises that the game will end here.
Should you stick with your door, or choose the other?
Boy or Girl
Consider a random family with two children. Given that one of the children is a boy, what is the probability that both children are boys?
Other variants:
- Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
- Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
- You are going to meet the king. Along the way, you are told that he has one sibling. What do you estimate is the probability that his sibling is a sister?
This video is somewhat correct, and somewhat incorrect…
Bertrand’s Boxes
Three boxes:
- Box A: 2 gold coins
- Box B: 2 silver coins
- Box C: 1 gold, 1 silver
A coin is drawn at random from one of the boxes. It is gold. What is the probability that the other coin in the same box as the drawn coin is gold?
Two Envelopes
You are given two envelopes of money, and promised that one of them contains twice as much money as the other. You will be allowed to open and keep the money in just one of them.
Argument:
- You choose one, say ‘A’. Then you are offered the choice to switch to the other. Should you?
- Let
be the amount of money in the envelope you chose first. - The other envelop contains
with and contains with . - The expected value of money in the other envelop is therefore
. - So you should switch.
What is wrong with this argument?