To determine whether being under the age of 12 is independent of visiting the inventions exhibit, we need to verify if the joint probability of both events occurring is equal to the product of their individual probabilities. Here’s the step-by-step process:
1. Given Probabilities:
- The probability of a visitor being under the age of 12, [tex]\( P(\text{under 12}) = 0.368 \)[/tex].
- The probability of a visitor deciding to visit the inventions exhibit, [tex]\( P(\text{visit exhibit}) = 0.321 \)[/tex].
- The joint probability of a visitor being under 12 and visiting the inventions exhibit, [tex]\( P(\text{under 12 and visit}) = 0.118 \)[/tex].
2. Calculate the Product of the Individual Probabilities:
To check for independence, we need to calculate the product of the individual probabilities of being under 12 and visiting the exhibit:
[tex]\[
P(\text{under 12}) \times P(\text{visit exhibit}) = 0.368 \times 0.321 = 0.118128
\][/tex]
3. Compare the Joint Probability with the Product of Individual Probabilities:
- The calculated product of the individual probabilities is [tex]\( 0.118128 \)[/tex].
- The given joint probability is [tex]\( 0.118 \)[/tex].
4. Conclusion:
- Since [tex]\( 0.118 \neq 0.118128 \)[/tex], the product of the individual probabilities does not equal the joint probability.
Hence, we conclude that being under the age of 12 is not independent of visiting the inventions exhibit at this museum. The correct justification is:
No, because 0.118 [tex]\( \neq \)[/tex] (0.368)(0.321)
