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18 Aug 2025, 10:46
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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Question Stats:
100% (04:06) correct
0% (00:00) wrong based on 1 sessions
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p is the probability that a school’s hockey team wins today. s is the independent probability that the school’s lacrosse team wins today. p and s are not both equal to 0.
Quantity A
p+s
Quantity B
ps
Quantity A
p+s
Quantity B
ps
30 Aug 2025, 03:55
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Official Explanation
The probability of any event is always between zero and one, inclusive, which means that \(0 ≤ p ≤ 1,\) and \(0 ≤ s ≤ 1\). Recall that the product of two decimals between 0 and 1 is smaller than either one of them (“fraction of a fraction”) and that the sum of any two positive numbers is greater than either one of them. This suggests that Quantity (A), p + s, is greater than Quantity (B), ps.
Choice (A) is correct, but don’t forget to check the boundary cases, because p or s could equal 0 or 1, when some of these general rules do not apply. If p equals 1, then p + s equals 1 + s, while ps equals 1s. Since \(1 + s > s,\) (A) is greater. If p = 0, then \(p + s = s,\) while \(ps = 0,\) and again (A) is greater.
*Pick plausible values for p and s, consistent with the fact that p and s represent probabilities, and test each pair. For example, if and then p + s equals 1, while ps equals and (A) is greater. If p = .1 and s = .1, then p + s equals .2, which is much greater than ps, or .01. Finally, if p = s = .9 or .8, then p + s > 1, while ps < 1, confirming that Quantity (A) is greater. By testing a wide range of possible values for p and s, we are fairly certain that (A) will always be greater.
Answer: A
The probability of any event is always between zero and one, inclusive, which means that \(0 ≤ p ≤ 1,\) and \(0 ≤ s ≤ 1\). Recall that the product of two decimals between 0 and 1 is smaller than either one of them (“fraction of a fraction”) and that the sum of any two positive numbers is greater than either one of them. This suggests that Quantity (A), p + s, is greater than Quantity (B), ps.
Choice (A) is correct, but don’t forget to check the boundary cases, because p or s could equal 0 or 1, when some of these general rules do not apply. If p equals 1, then p + s equals 1 + s, while ps equals 1s. Since \(1 + s > s,\) (A) is greater. If p = 0, then \(p + s = s,\) while \(ps = 0,\) and again (A) is greater.
*Pick plausible values for p and s, consistent with the fact that p and s represent probabilities, and test each pair. For example, if and then p + s equals 1, while ps equals and (A) is greater. If p = .1 and s = .1, then p + s equals .2, which is much greater than ps, or .01. Finally, if p = s = .9 or .8, then p + s > 1, while ps < 1, confirming that Quantity (A) is greater. By testing a wide range of possible values for p and s, we are fairly certain that (A) will always be greater.
Answer: A
