Let's tackle each part of this problem step by step.

Part (A): Without Repeats (Distinct Brands)

1. Total Possible Selections: Since 4 distinct brands are chosen from a total of 12 without repetition, there are $\binom{12}{4}$ ways to choose 4 brands out of 12. The number of ways to arrange these 4 brands correctly is $4! = 24$.

2. Successful Outcomes: For the consumer to correctly guess all 4 brands in the exact order, there’s only 1 successful way (the correct sequence of those 4 brands).

3. Probability Calculation: Therefore, the probability of correctly identifying all 4 brands by guessing is: $P(\text{correct without repeats}) = \frac{1}{\binom{12}{4} \cdot 4!}$

Let's calculate this.

$\binom{12}{4} = \frac{12!}{4!(12-4)!} = 495$

Then, the total number of arrangements of any 4 brands selected from 12 is: $495 \times 24 = 11880$

So, the probability of guessing all 4 brands correctly is: $P(\text{correct without repeats}) = \frac{1}{11880}$

Part (B): With Repeats Allowed

1. Total Possible Selections: Now, if repeats are allowed, there are $12$ choices for each of the 4 positions, so the total possible combinations of brands for the 4 slots is $12^4$.

2. Successful Outcomes: Again, there’s only 1 successful combination where the consumer correctly identifies all 4 brands in the correct order.

3. Probability Calculation: Therefore, the probability of correctly identifying all 4 brands by guessing is: $P(\text{correct with repeats}) = \frac{1}{12^4}$

Calculating $12^4$: $12^4 = 20736$

Thus, the probability of guessing all 4 brands correctly with repeats allowed is: $P(\text{correct with repeats}) = \frac{1}{20736}$

Summary of Answers

• (A): The probability of identifying all 4 brands correctly without repeats is $\frac{1}{11880}$.
• (B): The probability of identifying all 4 brands correctly with repeats allowed is $\frac{1}{20736}$.

Would you like more details on any of these steps?

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Related Questions:

1. How would the probability change if we chose 3 brands instead of 4?
2. What if there were only 10 brands to choose from instead of 12?
3. How does allowing repetition in answers affect probability outcomes in general?
4. Could we use a similar approach if the consumer had to match flavors instead of brands?
5. How would the probability change if we used a scoring system instead of requiring all correct guesses?

Tip: When calculating probabilities with combinations and arrangements, it's helpful to determine whether repetition is allowed, as it significantly changes the outcome.