on a tv game show,you try to win a prize that is hidden behind one of the three doors.after you choose a door, but before it is open,the host opens one of the other doors,behind which there is no prize.you can then switch the remaining closed door or stay with your orginial prize.
A: find the experimental probability of winning if your stragety is to stay with your orginal choice( hint:simulate by using one marked index card and two unmarked index cards.
B: find the experimental probability of winning if your stragety is to switch to the other door.
C: should you stay or switch in this game. Explain??
please help me i am confused...
thank you :)
Your problem is the classic "Monte Hall" problem
Wikepedia gives an excellent solution
http://en.wikipedia.org/wiki/Monty_Hall_problem
To find the experimental probability in this scenario, we can simulate the game using index cards. Remember, there is one prize and two non-prize cards.
A: If your strategy is to stay with your original choice, follow these steps:
1. Take three index cards - mark one card as the "prize" and leave the other two blank.
2. Shuffle the three cards, ensuring you do not know the arrangement.
3. Choose one card as your "guess" without looking at the face of the card.
4. Before revealing the card, the host opens one of the other two cards that you did not choose.
5. Calculate and record the number of times your original guess was the prize.
6. Repeat the process a sufficient number of times, ideally at least 100, to gather enough data.
7. The experimental probability of winning by staying with your original choice is the number of times your original guess was the prize divided by the total number of trials.
B: If your strategy is to switch to the other door, here are the steps:
1. Follow the same steps as in part A until you have chosen your card as the "guess."
2. Before revealing the card, the host must open one of the other two cards that you did not choose.
3. Instead of staying with your original choice, switch your guess to the remaining unopened door.
4. Calculate and record the number of times your switched guess was the prize.
5. Repeat the process a sufficient number of times, ideally at least 100.
6. The experimental probability of winning by switching to the other door is the number of times your switched guess was the prize divided by the total number of trials.
C: To determine whether you should stay or switch, compare the experimental probabilities from parts A and B.
If the experimental probability of winning by staying (part A) is higher than the probability of winning by switching (part B), then you should stay with your original choice. If the experimental probability of winning by switching (part B) is higher than the probability of winning by staying (part A), then you should switch.
The explanation behind this is that when you initially choose a door, you have a 1/3 chance of picking the correct door. When the host opens one of the other doors, it does not affect the probability of your original choice being correct. However, by switching, you are essentially betting on the probability that your initial choice was wrong, which gives you a 2/3 chance of winning.