Certainly! Let's analyze each statement in detail.
1. Statement: [tex]\( P(6) = P(1) \)[/tex]
Since each slip of paper has an equal probability of being drawn and there are no distinguishing factors among the slips of paper, the probability of drawing any specific number is the same. Therefore, the probability of drawing 6 is equal to the probability of drawing 1.
True.
2. Statement: [tex]\( P(5) = \frac{1}{2} \)[/tex]
To find the probability of drawing the slip labeled with the number 5, we consider that there are 10 slips of paper. The probability of drawing any one specific slip is [tex]\( \frac{1}{10} \)[/tex]. The probability [tex]\( \frac{1}{2} \)[/tex] would be correct only if there were two slips in total, and one was labeled 5. Therefore, this statement is incorrect.
False.
3. Statement: [tex]\( P(>10) = 0 \)[/tex]
Given that the set of numbers on the slips is [tex]\(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)[/tex], there are no numbers greater than 10. So, the probability of drawing a number greater than 10 is zero.
True.
4. Statement: [tex]\( P(1 < x < 10) = 100\% \)[/tex]
This statement claims that the probability of drawing a number [tex]\( x \)[/tex] where [tex]\( 1 < x < 10 \)[/tex] (i.e., only numbers from 2 to 9) is 100%. However, there are slips labeled 1 and 10 as well, which means it's possible to draw these numbers. Thus, the probability of [tex]\( 1 < x < 10 \)[/tex] is not 100%.
False.
5. Statement: [tex]\( S = \{1,2,3,4,5,6,7,8,9,10\} \)[/tex]
The set [tex]\( S \)[/tex] indeed consists of the numbers 1 through 10. Therefore, this statement is correct.
True.
6. Statement: If [tex]\( A \subset S \)[/tex]; [tex]\( A \)[/tex] could be [tex]\(\{1, 3, 5, 7, 9\}\)[/tex]
A subset [tex]\( A \)[/tex] of a set [tex]\( S \)[/tex] can be any combination of elements from [tex]\( S \)[/tex]. The subset [tex]\( \{1, 3, 5, 7, 9\} \)[/tex] is a valid subset of the set [tex]\( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]. Therefore, this statement is correct.
True.
To summarize, the true statements are:
- [tex]\( P(6) = P(1) \)[/tex]
- [tex]\( P(>10) = 0 \)[/tex]
- [tex]\( S = \{1,2,3,4,5,6,7,8,9,10\} \)[/tex]
- If [tex]\( A \subset S \)[/tex]; [tex]\( A \)[/tex] could be [tex]\(\{1, 3, 5, 7, 9\}\)[/tex]
And the false statements are:
- [tex]\( P(5) = \frac{1}{2} \)[/tex]
- [tex]\( P(1 < x < 10) = 100\%\)[/tex]
