Question

based on the percentages in the tree diagram, is the new test a reliable indicator that a child with a positive test actually has a food allergy? use the drop-down menus to explain. actually has allergy? yes 8% no 92% test shows allergy? yes 90% no 10% (for yes allergy) yes 15% no 85% (for no allergy) click the arrows to choose an answer from each menu. the probability that a child will receive a positive test is choose... . for a child with a positive test, the probability that the child actually has a food allergy is approximately choose... . the new test choose... likely to be a reliable indicator that the child has a food allergy.

Explanation:

Step1: Calculate P(Positive Test)

To find the probability of a positive test, we consider two cases: child has allergy (8% chance) and tests positive (90% of those cases), and child does not have allergy (92% chance) and tests positive (15% of those cases).
Using the law of total probability:
\( P(\text{Positive}) = P(\text{Allergy}) \times P(\text{Positive}|\text{Allergy}) + P(\text{No Allergy}) \times P(\text{Positive}|\text{No Allergy}) \)
Substitute values:
\( P(\text{Positive}) = 0.08 \times 0.90 + 0.92 \times 0.15 \)
\( = 0.072 + 0.138 = 0.21 \) (or 21%).

Step2: Calculate P(Allergy|Positive)

Using Bayes' theorem:
\( P(\text{Allergy}|\text{Positive}) = \frac{P(\text{Positive}|\text{Allergy}) \times P(\text{Allergy})}{P(\text{Positive})} \)
We know \( P(\text{Positive}|\text{Allergy}) = 0.90 \), \( P(\text{Allergy}) = 0.08 \), and \( P(\text{Positive}) = 0.21 \) from Step 1.
Substitute:
\( P(\text{Allergy}|\text{Positive}) = \frac{0.90 \times 0.08}{0.21} = \frac{0.072}{0.21} \approx 0.3429 \) (or ~34.3%).

Step3: Evaluate Reliability

Since the conditional probability \( P(\text{Allergy}|\text{Positive}) \approx 34.3\% \) is not very high (less than 50%), the test is not very reliable.

Answer:

The probability that a child will receive a positive test is \( \boldsymbol{21\%} \) (or 0.21). For a child with a positive test, the probability that the child actually has a food allergy is approximately \( \boldsymbol{34.3\%} \) (or ~0.34). The new test is \( \boldsymbol{\text{not very}} \) likely to be a reliable indicator that the child has a food allergy.