Question

a survey was conducted at a local ballroom dance studio asking students if they had ever competed in the following dance categories: - smooth - rhythm - standard the results were then presented to the owner in the following venn diagram. if a student is chosen at random, what is the probability that: write your answers in percent form. round to the nearest tenth of a percent. a) the student has competed in none of the categories? % b) the student has competed in all three of these categories? % c) the student has competed in smooth or standard, but not rhythm? % d) the student has competed in rhythm and standard, but not smooth? % e) the student has competed in rhythm. %

Explanation:

Step1: Calculate total number of students

Add all the numbers in the Venn - diagram and the number outside the Venn - diagram (assume the number outside is 7). Total students \(N=11 + 13+5 + 6+9+12 + 4+7=67\).

Step2: Probability that student has competed in none of the categories

The number of students who have competed in none of the categories is 7. Probability \(P(a)=\frac{7}{67}\approx0.1045\). In percent form, \(P(a)=10.5\%\).

Step3: Probability that student has competed in all three categories

The number of students who have competed in all three categories is 6. Probability \(P(b)=\frac{6}{67}\approx0.0896\). In percent form, \(P(b)=9.0\%\).

Step4: Probability that student has competed in Smooth or Standard, but not Rhythm

The number of students who have competed in Smooth or Standard, but not Rhythm is \(11 + 4=15\). Probability \(P(c)=\frac{15}{67}\approx0.2239\). In percent form, \(P(c)=22.4\%\).

Step5: Probability that student has competed in Rhythm and Standard, but not Smooth

The number of students who have competed in Rhythm and Standard, but not Smooth is 12. Probability \(P(d)=\frac{12}{67}\approx0.1791\). In percent form, \(P(d)=17.9\%\).

Step6: Probability that student has competed in Rhythm

The number of students who have competed in Rhythm is \(13 + 5+6 + 12=36\). Probability \(P(e)=\frac{36}{67}\approx0.5373\). In percent form, \(P(e)=53.7\%\).

Answer:

a) \(10.5\%\)
b) \(9.0\%\)
c) \(22.4\%\)
d) \(17.9\%\)
e) \(53.7\%\)