Question

lara surveyed all of the 11th grade students at her school about the number of languages they speak and whether or not they have allergies. here are her results. complete the table giving the conditional distribution of number of languages spoken for students with no allergies. round your answers to the nearest tenth of a percent. 1 language 2 languages 3 languages 6 languages

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{n(A\cap B)}{n(B)}\). In the context of the table, for the number of languages \(x\) (where \(x = 1,2,3,6\)) and the event of having no allergies, the formula for the percentage is \(\text{Percentage}=\frac{\text{Number of students with }x\text{ languages and no allergies}}{\text{Total number of students with no allergies}}\times100\). The total number of students with no allergies \(n = 44\).

Step2: Calculate for \(x = 1\) language

For \(x = 1\) language, the number of students with 1 language and no allergies is \(28\). Using the formula \(\text{Percentage}=\frac{28}{44}\times100=\frac{2800}{44}\approx63.6\%\).

Step3: Calculate for \(x = 2\) languages

For \(x = 2\) languages, the number of students with 2 languages and no allergies is \(14\). Using the formula \(\text{Percentage}=\frac{14}{44}\times100=\frac{1400}{44}\approx31.8\%\).

Step4: Calculate for \(x = 3\) languages

For \(x = 3\) languages, the number of students with 3 languages and no allergies is \(1\). Using the formula \(\text{Percentage}=\frac{1}{44}\times100=\frac{100}{44}\approx2.3\%\).

Step5: Calculate for \(x = 6\) languages

For \(x = 6\) languages, the number of students with 6 languages and no allergies is \(1\). Using the formula \(\text{Percentage}=\frac{1}{44}\times100=\frac{100}{44}\approx2.3\%\).

Answer:

1 language: \(63.6\%\), 2 languages: \(31.8\%\), 3 languages: \(2.3\%\), 6 languages: \(2.3\%\)