montyhallstuff
A sample space is the set of all possible outcomes of a random experiment. An event is a subset of a sample space.
Let S be the finite sample space with equally likely outcomes in it. Let E be an event. Then the probability of E is the number of outcomes in E over the total number of outcomes in S.
Now consider 52 cards in a deck. The sample space of outcomes here are the 52 cards. What is the event that the chosen card is a black face card? E = {J-club,Q-club,K-club,J-spade,Q-spade,K-spade}. What is the probability that the chosen card is a black face card? 6/52 by the formula above.
Monty Hall problem can be solved by using the above as a guide.
Sample space here : {switch, switch, stay-with-your-original-choice}. By the formula above, the probability of switching is 2/3 and the probability of staying put is 1/3.
If you're doubting the formula, I can provide you with a proof.
Right before any of the doors is opened we have no idea what to expect, so there are 3 equaly likely possibilities:
1. prize is behind A 2. prize is behind B 3. prize is behind C.
In case (1) you win by staying with your choice of A. In case (2) Monty Hall must open C (by the rules), so you'll win if you switch to B and in (3) the host will open B(by the rules) and you should switch to C.
In 2 cases out of 3 (2/3) you win by switching and in only one case out of 3 (1/3) you win by staying with your first choice. Since 2/3 > 1/3, you should switch.
This problem is easy if you know the defintions of sample space, event and the Equally Likely Probability Formula.