Answer :
To solve this problem, let's break it down step by step.
1. Identify the coordinates and lengths of segments:
- The entire line segment [tex]\( JM \)[/tex] has endpoints at 0 and 25. Hence, the total length of [tex]\( JM \)[/tex] is:
[tex]\[ JM = 25 - 0 = 25 \][/tex]
- Points [tex]\( K \)[/tex] and [tex]\( L \)[/tex] have coordinates 5 and 12, respectively. Thus, the length of segment [tex]\( JL \)[/tex] is:
[tex]\[ JL = 12 - 0 = 12 \][/tex]
- The length of segment [tex]\( KL \)[/tex] (since [tex]\( K \)[/tex] is at 5 and [tex]\( L \)[/tex] is at 12) is:
[tex]\[ KL = 12 - 5 = 7 \][/tex]
2. Calculate probabilities:
- The probability that a point placed on [tex]\( JM \)[/tex] falls within segment [tex]\( JL \)[/tex] is the length of [tex]\( JL \)[/tex] over the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of JL} = \frac{JL}{JM} = \frac{12}{25} = 0.48 \][/tex]
- The probability that a second point falls outside the segment [tex]\( KL \)[/tex] (i.e., on [tex]\( JM \)[/tex] but not on [tex]\( KL \)[/tex]) can be found by subtracting the length of [tex]\( KL \)[/tex] from the total length of [tex]\( JM \)[/tex], and then dividing by the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of not KL} = \frac{JM - KL}{JM} = \frac{25 - 7}{25} = \frac{18}{25} = 0.72 \][/tex]
3. Calculate combined probability:
- The combined probability that the first point is placed on [tex]\( JL \)[/tex] and the second point is not placed on [tex]\( KL \)[/tex] is the product of the two individual probabilities:
[tex]\[ \text{Combined probability} = 0.48 \times 0.72 = 0.3456 \][/tex]
4. Convert probability to a fraction:
- To express this probability as a fraction of the total possible number segments (since [tex]\( JM \)[/tex] is discretized), multiply by [tex]\( 25^2 \)[/tex] (the square of the total length, as we're considering two points on the segment):
[tex]\[ \text{Fraction solution: } 0.3456 \times 25^25 = 0.3456 \times 625 = 216 \][/tex]
Thus, the probability that a point on [tex]\( JM \)[/tex] is placed first on [tex]\( JL \)[/tex] and a second point is not placed on [tex]\( KL \)[/tex] is:
[tex]\[ \frac{216}{625} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{216}{625}} \][/tex]
1. Identify the coordinates and lengths of segments:
- The entire line segment [tex]\( JM \)[/tex] has endpoints at 0 and 25. Hence, the total length of [tex]\( JM \)[/tex] is:
[tex]\[ JM = 25 - 0 = 25 \][/tex]
- Points [tex]\( K \)[/tex] and [tex]\( L \)[/tex] have coordinates 5 and 12, respectively. Thus, the length of segment [tex]\( JL \)[/tex] is:
[tex]\[ JL = 12 - 0 = 12 \][/tex]
- The length of segment [tex]\( KL \)[/tex] (since [tex]\( K \)[/tex] is at 5 and [tex]\( L \)[/tex] is at 12) is:
[tex]\[ KL = 12 - 5 = 7 \][/tex]
2. Calculate probabilities:
- The probability that a point placed on [tex]\( JM \)[/tex] falls within segment [tex]\( JL \)[/tex] is the length of [tex]\( JL \)[/tex] over the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of JL} = \frac{JL}{JM} = \frac{12}{25} = 0.48 \][/tex]
- The probability that a second point falls outside the segment [tex]\( KL \)[/tex] (i.e., on [tex]\( JM \)[/tex] but not on [tex]\( KL \)[/tex]) can be found by subtracting the length of [tex]\( KL \)[/tex] from the total length of [tex]\( JM \)[/tex], and then dividing by the total length of [tex]\( JM \)[/tex]:
[tex]\[ \text{Probability of not KL} = \frac{JM - KL}{JM} = \frac{25 - 7}{25} = \frac{18}{25} = 0.72 \][/tex]
3. Calculate combined probability:
- The combined probability that the first point is placed on [tex]\( JL \)[/tex] and the second point is not placed on [tex]\( KL \)[/tex] is the product of the two individual probabilities:
[tex]\[ \text{Combined probability} = 0.48 \times 0.72 = 0.3456 \][/tex]
4. Convert probability to a fraction:
- To express this probability as a fraction of the total possible number segments (since [tex]\( JM \)[/tex] is discretized), multiply by [tex]\( 25^2 \)[/tex] (the square of the total length, as we're considering two points on the segment):
[tex]\[ \text{Fraction solution: } 0.3456 \times 25^25 = 0.3456 \times 625 = 216 \][/tex]
Thus, the probability that a point on [tex]\( JM \)[/tex] is placed first on [tex]\( JL \)[/tex] and a second point is not placed on [tex]\( KL \)[/tex] is:
[tex]\[ \frac{216}{625} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{216}{625}} \][/tex]
