Problem 79 - Basic Statistics for Business and Economics with Student

Question

During the 1970 s the qame show Let's Make a Deal had a long run on TV. In the show a contestant was given a choice of three doors, behind one of which was a prize. The other two doors contained a gag gift of some type. After the contestant selected a door, the host of the show then revealed to them one of the doors from among the two not selected. The host asked the contestant if they wished to switch doors to one of those not chosen. Should the contestant switch? Are the odds of winning increased by switching doors?

Step-by-Step Solution

Answer

Yes, the contestant should switch; this increases the chances of winning from \( \frac{1}{3} \) to \( \frac{2}{3} \).

Step 1: Initial Probability Assessment

Initially, the contestant picks one of the three doors. The probability that the prize is behind the selected door is \( \frac{1}{3} \). Conversely, this means there is a \( \frac{2}{3} \) chance that the prize is behind one of the other two doors.

Step 2: Host's Action

The host, who knows where the prize is, opens one of the two doors not selected by the contestant and reveals a gag gift. Thus, there is still a \( \frac{2}{3} \) chance that the prize is behind the unchosen doors (the door the contestant did not initially select and the host did not open).

Step 3: Probability After Switching

If the contestant switches, they will choose the only other unopened door, which still holds the \( \frac{2}{3} \) probability of containing the prize.

Step 4: Conclusion

Therefore, by switching, the contestant improves their chances of winning from \( \frac{1}{3} \) to \( \frac{2}{3} \). The odds of winning are increased by switching to the other door.

Key Concepts

Understanding Conditional Probability

Conditional probability is a fundamental concept in probability that deals with finding the probability of an event occurring given that another event has already occurred. In simpler terms, it allows us to narrow down the likelihood of an event by taking into account additional, relevant information.
For instance, when you know certain facts about the situation or have observed certain outcomes, conditional probability allows you to update the probabilities accordingly. In the case of the Monty Hall problem, conditional probability plays an important role. Initially, you might think each door has a one-third chance of hiding the prize. However, once the host reveals a door with a gag gift, the situation changes.
This newly given information modifies the probability because it involves the host's choice. Now, we know there’s a higher probability attached to the door not initially picked by the contestant.
In mathematical terms, the conditional probability can be represented as \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)where \( P(A \mid B) \) is the probability of event A given event B has occurred. Understanding this can significantly help in many decision-making scenarios where you have additional information at hand.

Decoding the Monty Hall Problem

The Monty Hall problem is a classic example of probability puzzles, which at first sight appear counter-intuitive but when solved, unlock the magic behind probability theory.
The problem comes from a television game show setting where there's an initial scenario of uncertainty followed by a change due to an intervention—in this case, the host's action.
Let’s break down the puzzle:

• The initial choice has a probability of \( \frac{1}{3} \) of picking the prize-bearing door.
• The host, who knows where the prize is, opens a door without the prize.
• This action changes the dynamics since it gives additional information narrowing down the options.

Upon the host's action, the probability of the prize being behind the remaining unopened door rises to \( \frac{2}{3} \).
Therefore, switching is statistically more favorable and doubles the probability of winning. Learning from the Monty Hall problem can also enhance your understanding of how additional information can affect likelihood in everyday decision making.

Strategies in Decision Making

Making decisions, especially under uncertainty, is something we do almost daily. The Monty Hall problem is a great illustration of how sound strategies and understanding probabilities can influence outcomes.
Here are some insights on decision-making that naturally flow from understanding the Monty Hall problem:

• Use additional information wisely: Don't ignore new information. It can significantly change the probabilities and thus the best course of action.
• Assess initial probabilities: Always start by understanding the situation and setting initial probabilities.
• Be open to change: Just like in the Monty Hall problem, sometimes the best decision is to change your initial choice based on new data.

By applying these strategies, whether in games, investments, or daily life choices, you make informed decisions that maximize your chance of success.
By coupling understanding with practice, just like in the Monty Hall problem, effective decision-making can achieve better results reliably.