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Conditional probability: the following problem is loosely based on the television game show Let's Make a Deal. At the end of the show, a contestant is asked to choose one of three large boxes, where one box contains a fabulous prize and the other two boxes contain lesser prizes. After the contestant chooses a box, Monty Hall, the host of the show, opens one of the two boxes containing smaller prizes. (In order to keep the conclusion suspenseful, Monty does not open the box selected by the contestant.) Monty offers the contestant the opportunity to switch from the rhosen box to the remaining unopened box. Should the contestant switch or stay with the original choice? Calculate the probability that the contestant wins under each strategy. This is an exercise in being clear about the information that should be conditioned on when constructing a probability judgment. See Selvin (1975) and Morgan et al. (1991) for further discussion of this problem.

Step by step solution

01

Initial Setup

There are three boxes: Box A, Box B, and Box C. One contains a fabulous prize (F), while the others contain lesser prizes (L). The contestant chooses one box, say Box A.

02

Monty's Choice

Monty Hall will always open a box with a lesser prize. Therefore, if the contestant initially chose Box A, Monty will open either Box B or Box C, whichever contains a lesser prize, revealing it.

03

Probability of Winning by Sticking

If the contestant sticks with Box A, the probability that Box A contains the fabulous prize (F) was originally 1 out of 3, since this was a random choice from the three boxes:\[P(\text{Win by sticking}) = \frac{1}{3}\]

04

Probability of Winning by Switching

If the contestant switches after Monty opens one box, the probability that the other unopened box contains the fabulous prize is 2 out of 3. This is because:- If Box A was a lesser prize (which happens 2 out of 3 times), Monty's opened box also has a lesser prize, and the remaining unopened box must have the fabulous prize.- So, switching gives a probability of:\[P(\text{Win by switching}) = \frac{2}{3}\]

05

Conclusion

Comparing the probabilities, switching offers a higher chance (\(\frac{2}{3}\)) of winning the fabulous prize, whereas sticking with the initial choice offers a lower probability of \(\frac{1}{3}\). Therefore, the contestant should switch to increase their chances of winning.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Judgment

Probability judgment involves evaluating the chances of different outcomes, based on available information. It's like using clues to make a guess about what might happen. Imagine you're in a game show with three boxes.

• One box has a fabulous prize and the other two have less exciting ones.
• You pick one box, but you don't know where the prize is.
• The host, Monty Hall, then opens one of the two boxes you didn't choose and shows you a lesser prize.

Now you have more information than before. Probability judgment helps you decide whether to stick with your initial choice or switch to the other unopened box. You'd think the chances are equal between two boxes, right? But no! With smart probability judgment, you'll find that switching doubles your chances of winning. It's about making sense of new information and adjusting your decision to maximize your winning chance.

Game Theory

Game theory is like a roadmap for making strategic decisions in situations where your outcome also depends on others' actions. It's a big part of understanding games like the Monty Hall problem. Who are the players?

• There's you, the contestant, who picks a box hoping it holds the prize.
• Then there's Monty Hall, a host who knows where the fabulous prize is and helps in decision-making by revealing a lesser prize from the boxes you didn’t pick.

In this scenario, game theory helps you realize that while initially your choice might seem random, Monty's actions actually give you more information. After Monty opens a box with a lesser prize, game theory shows that switching boxes gives you a higher chance of winning the fabulous prize, specifically a 2 out of 3 probability. Using game theory, you perform actions based not just on your own choice but considering the host's actions as well.

Monty Hall Problem

The Monty Hall problem is a fascinating probability puzzle that challenges our intuitive judgments. At first glance, it might seem simple: pick a box, but the twists make it intriguing and educational. Let's dive into how it unfolds:

• You start by picking one of three boxes. Initially, your chance of picking the box with the prize is \( \frac{1}{3} \).
• Monty Hall, who knows where the prize is, will then open one of the remaining boxes, which he knows is a dud. He never opens the box with the prize, giving you a mighty clue.
• Now, you're faced with a choice: stick with your first box or switch to the other unopened box.

Here's where logic and math come in. If you stick with your original choice, the chance of winning is \( \frac{1}{3} \). But if you switch, your chances soar to \( \frac{2}{3} \). That's because Monty gives you extra information by showing that one box is absolutely not the winner. This counter-intuitive solution is what makes the Monty Hall problem a great way to explore conditional probabilities and challenge our initial instincts.