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\(\mathrm{M} = \{-6, -5, -4, -3, -2\}\)
\(\mathrm{T} = \{-2, -1, 0, 1, 2, 3\}\)
If an integer is to be randomly selected from set M above and an integer is to be randomly selected from set T above, what is the probability that the product of the two integers will be negative?
Let's start by understanding what we're looking for in everyday terms. We have two bags of numbers: bag M with negative numbers \(\{-6, -5, -4, -3, -2\}\), and bag T with a mix of negative, zero, and positive numbers \(\{-2, -1, 0, 1, 2, 3\}\). We're going to pick one number from each bag and multiply them together.
The question asks: what's the probability that this product will be negative?
Think about multiplication in simple terms: when do we get a negative result? The key insight is that we get a negative product when we multiply numbers with opposite signs - that is, one positive number times one negative number gives us a negative result.
So we need to find the probability of selecting numbers with opposite signs from our two sets.
Process Skill: TRANSLATE - Converting the probability question into a clear mathematical requirement about sign patterns
Let's examine each set and categorize the numbers by their signs:
Set M = \(\{-6, -5, -4, -3, -2\}\)
Set T = \(\{-2, -1, 0, 1, 2, 3\}\)
Now, remember that for a product to be negative, we need opposite signs. Since all numbers in M are negative, we need to pick a positive number from T to get a negative product.
Let's count the total possible outcomes first, then count the favorable ones.
Total possible outcomes: We can pick any of the 5 numbers from M and any of the 6 numbers from T. So total outcomes = \(5 \times 6 = 30\).
Now for favorable outcomes (negative products):
Since every number in M is negative, we get a negative product when we pair any number from M with a positive number from T.
Note: If we pick 0 from T, the product would be 0 (neither positive nor negative). If we pick a negative number from T, we'd get negative × negative = positive.
Process Skill: CONSIDER ALL CASES - Systematically accounting for all sign combinations
Probability is simply the ratio of favorable outcomes to total possible outcomes.
Probability = Favorable outcomes / Total outcomes = \(\frac{15}{30} = \frac{1}{2}\)
Let's verify this makes sense: since all numbers in M are negative, and exactly half the numbers in T are positive (3 out of 6), we'd expect the probability to be \(\frac{1}{2}\).
The probability that the product of two randomly selected integers (one from each set) will be negative is \(\frac{1}{2}\).
This matches answer choice D.
Verification: 15 favorable outcomes out of 30 total outcomes gives us \(\frac{15}{30} = \frac{1}{2}\), confirming our answer.
1. Misunderstanding when products are negative
Students often confuse the rules for when multiplication results in negative numbers. They might think that multiplying two negative numbers gives a negative result, when actually negative × negative = positive. The key insight that products are negative only when signs are opposite (positive × negative OR negative × positive) is frequently missed.
2. Overlooking the impact of zero in set T
Many students fail to recognize that zero creates a special case. When zero is multiplied by any number, the result is zero, which is neither positive nor negative. Students might incorrectly count zero-containing pairs as either favorable or unfavorable outcomes, rather than excluding them from the "negative product" count.
3. Misinterpreting "randomly selected" as meaning equal probability from each set
Some students might think they need to find equal numbers from each set or that the sets should be the same size for the probability calculation. They may get confused about how to handle sets of different sizes (M has 5 elements, T has 6 elements) in probability calculations.
1. Incorrect counting of sign categories
Students frequently miscount the numbers in each category. For set T = \(\{-2, -1, 0, 1, 2, 3\}\), they might count 2 negatives, 1 zero, and 3 positives incorrectly, or forget to account for zero as a separate category that doesn't contribute to negative products.
2. Arithmetic errors in calculating total outcomes
When calculating total possible outcomes as \(5 \times 6 = 30\), students may make simple multiplication errors or forget that they need to multiply the sizes of both sets. Alternatively, they might add instead of multiply (\(5 + 6 = 11\)), confusing this with other probability scenarios.
3. Wrong pairing for favorable outcomes
Students might incorrectly calculate favorable outcomes by pairing negatives from M with negatives from T (giving \(5 \times 2 = 10\)) instead of pairing negatives from M with positives from T (giving \(5 \times 3 = 15\)). This stems from the conceptual error about when products are negative.
1. Fraction simplification errors
After correctly calculating \(\frac{15}{30}\), students might make errors in simplifying the fraction. They could get confused with the arithmetic and select \(\frac{3}{5}\) (thinking \(\frac{15}{30} = \frac{15}{30}\)) or make other simplification mistakes instead of recognizing that \(\frac{15}{30} = \frac{1}{2}\).
2. Selecting the complement probability
Students might calculate the probability of getting a positive or zero product instead of a negative product, then select that answer. Since the probability of a negative product is \(\frac{1}{2}\), the probability of a non-negative product is also \(\frac{1}{2}\), but they might confuse which one the question is asking for.