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The National Public Radio show Car Talk has a feature called "The Puzzler." Listeners are asked to send in answers to some puzzling questions-usually about cars but sometimes about probability (which, of course, must account for the incredible popularity of the program!). Suppose that for a car question, 800 answers are submitted, of which 50 are correct. a. Suppose that the hosts randomly select two answers from those submitted with replacement. Calculate the probability that both selected answers are correct. (For purposes of this problem, keep at least five digits to the right of the decimal.) b. Suppose now that the hosts select the answers at random but without replacement. Use conditional probability to evaluate the probability that both answers selected are correct. How does this probability compare to the one computed in Part (a)?

Step by step solution

01

Calculate the Probability with Replacement

The total number of answers submitted is 800, and the number of correct answers submitted is 50. The probability of selecting a correct answer on a given attempt - \(P(Correct)\) - is therefore \(\frac{50}{800}\), which simplifies to \(\frac{1}{16}\). When replacement is involved, the probabilities don't change with each attempt because the total number of answers remains the same. Therefore, the probability of getting two correct answers when picking with replacement is the product of the probabilities of picking a correct answer twice \((\frac{1}{16})^2\), which equals \(\frac{1}{256}\).

02

Calculate the Probability without Replacement

When picking without replacement, the pool of answers decreases by one each time an answer is selected. The probability of picking a correct answer on the first attempt - \(P(Correct1)\) - remains \(\frac{50}{800}\) or \(\frac{1}{16}\). However, the probability of picking a correct answer on the second attempt - \(P(Correct2 | Correct1)\) - is \(\frac{49}{799}\), as both the numerator and the denominator decrease by one. The probability of picking two correct answers without replacement is the product of these probabilities, \(\frac{1}{16} * \frac{49}{799} = \frac{49}{12784}\).

03

Compare the Two Probabilities

By calculating the decimal equivalent of both probabilities, we find that the probability with replacement is \(0.00391\) while the probability without replacement is \(0.00383\). Though they are close, it is slightly more likely to get two correct answers when replacing after each pick than when not replacing.

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Key Concepts

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

Conditional Probability

Conditional probability helps us determine the likelihood of an event occurring given a certain condition. In the car question, if the first answer selected is correct, the probability of the second answer also being correct changes depending on whether or not the first answer is replaced.
When dealing with events without replacement, the probability of the second event is dependent on the outcome of the first event. For instance, if you've already picked a correct answer once and do not replace it, there is one less correct answer available in the pool. This is why conditional probability is so essential to understand in these scenarios.

• Formula for conditional probability is: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
• "\(P(A|B)\)" reads as the probability of event A given event B has occurred.
• It adjusts the sample space based on prior outcomes.

Replacement vs. Non-Replacement

Replacement and non-replacement are critical concepts when dealing with probabilities, as they define whether the chosen item is placed back into the sample pool after being selected.

• With Replacement: The probability for each selection remains constant because the sample pool does not change. For example, if you select a correct answer and replace it, the next choice has the same probability of being correct.
• Without Replacement: The sample pool changes with each selection, meaning that the probabilities for subsequent selections are altered. When you pick a correct answer and do not replace it, there is one less correct answer available, decreasing the probability of picking another correct answer.

In our exercise, this distinction affects how we calculate the likelihood of getting two correct answers when choosing with or without replacement.

Calculating Probabilities

To calculate probabilities, we need to determine how likely an event is to happen relative to the total number of possible outcomes. This involves simple division of favorable outcomes by the total outcomes.
For example, in the exercise, we have 800 total answers and 50 correct answers. Thus, the probability of picking a correct answer on your first try is \( \frac{50}{800} = \frac{1}{16} \).

• With Replacement: The probability for both selections is the product of picking a correct answer each time, i.e., \( \left(\frac{1}{16}\right)^2 = \frac{1}{256} \).
• Without Replacement: The probability changes after the first pick. If the first answer is correct, only 49 correct answers remain out of 799 total, making the probability for the second correct pick \( \frac{49}{799} \). The combined probability then is \( \frac{1}{16} \times \frac{49}{799} = \frac{49}{12784} \).

Taking the decimal form helps in comparison, where we found the probability of both correct picks with replacement as 0.00391 and without replacement as 0.00383. This subtle difference is important for understanding how these two concepts affect outcome probabilities.