Answer :
Let's compare the probabilities given in the problem statement for each club:
- The probability of a student joining the Garden Club ([tex]\(P(\text{Garden})\)[/tex]) is 0.1.
- The probability of a student joining the Robotics Club ([tex]\(P(\text{Robotics})\)[/tex]) is 0.2.
- The probability of a student joining the Technology Club ([tex]\(P(\text{Technology})\)[/tex]) is 0.5.
- The probability of a student joining the Zoology Club ([tex]\(P(\text{Zoology})\)[/tex]) is 0.2.
Now, let's evaluate each statement individually:
1. The student will be more unlikely to join the Robotics Club than the Garden Club because [tex]\(P(\text{Robotics}) < P(\text{Garden})\)[/tex].
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(P(\text{Garden}) = 0.1\)[/tex]
- [tex]\(0.2 \not< 0.1\)[/tex] (This statement is false)
2. The student will be equally likely to join the Robotics Club or Technology Club because [tex]\(P(\text{Robotics}) = P(\text{Technology})\)[/tex].
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(P(\text{Technology}) = 0.5\)[/tex]
- [tex]\(0.2 \neq 0.5\)[/tex] (This statement is false)
3. The student will be more likely to join the Zoology Club than the Robotics Club because [tex]\(P(\text{Zoology}) > P(\text{Robotics})\)[/tex].
- [tex]\(P(\text{Zoology}) = 0.2\)[/tex]
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(0.2 \not> 0.2\)[/tex] (This statement is false)
4. The student will be more unlikely to join the Garden Club than the Robotics Club because [tex]\(P(\text{Garden}) < P(\text{Robotics})\)[/tex].
- [tex]\(P(\text{Garden}) = 0.1\)[/tex]
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(0.1 < 0.2\)[/tex] (This statement is true)
The correct interpretation is:
The student will be more unlikely to join the Garden Club than the Robotics Club because [tex]\(P(\text{Garden}) < P(\text{Robotics})\)[/tex].
So the true statement is:
"The student will be more unlikely to join the Garden Club than the Robotics Club because [tex]\(P(\text{Garden}) < P(\text{Robotics})\)[/tex]."
Therefore, the correct statement index is 4.
- The probability of a student joining the Garden Club ([tex]\(P(\text{Garden})\)[/tex]) is 0.1.
- The probability of a student joining the Robotics Club ([tex]\(P(\text{Robotics})\)[/tex]) is 0.2.
- The probability of a student joining the Technology Club ([tex]\(P(\text{Technology})\)[/tex]) is 0.5.
- The probability of a student joining the Zoology Club ([tex]\(P(\text{Zoology})\)[/tex]) is 0.2.
Now, let's evaluate each statement individually:
1. The student will be more unlikely to join the Robotics Club than the Garden Club because [tex]\(P(\text{Robotics}) < P(\text{Garden})\)[/tex].
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(P(\text{Garden}) = 0.1\)[/tex]
- [tex]\(0.2 \not< 0.1\)[/tex] (This statement is false)
2. The student will be equally likely to join the Robotics Club or Technology Club because [tex]\(P(\text{Robotics}) = P(\text{Technology})\)[/tex].
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(P(\text{Technology}) = 0.5\)[/tex]
- [tex]\(0.2 \neq 0.5\)[/tex] (This statement is false)
3. The student will be more likely to join the Zoology Club than the Robotics Club because [tex]\(P(\text{Zoology}) > P(\text{Robotics})\)[/tex].
- [tex]\(P(\text{Zoology}) = 0.2\)[/tex]
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(0.2 \not> 0.2\)[/tex] (This statement is false)
4. The student will be more unlikely to join the Garden Club than the Robotics Club because [tex]\(P(\text{Garden}) < P(\text{Robotics})\)[/tex].
- [tex]\(P(\text{Garden}) = 0.1\)[/tex]
- [tex]\(P(\text{Robotics}) = 0.2\)[/tex]
- [tex]\(0.1 < 0.2\)[/tex] (This statement is true)
The correct interpretation is:
The student will be more unlikely to join the Garden Club than the Robotics Club because [tex]\(P(\text{Garden}) < P(\text{Robotics})\)[/tex].
So the true statement is:
"The student will be more unlikely to join the Garden Club than the Robotics Club because [tex]\(P(\text{Garden}) < P(\text{Robotics})\)[/tex]."
Therefore, the correct statement index is 4.