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

Sum all the values in the Venn - diagram and the outside value. Let the number of students who competed in Smooth be \(S\), Rhythm be \(R\), and Standard be \(T\). The values in the Venn - diagram are: \(S\) only \(=15\), \(S\cap R = 7\), \(S\cap T=6\), \(R\) only \( = 8\), \(R\cap T = 3\), \(T\) only \(=6\), and the number of students outside the three circles \(=5\).
The total number of students \(N=15 + 7+6 + 8+3 + 6+5=50\).

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

The number of students who competed in none of the categories is \(5\). The probability \(P_1=\frac{5}{50}=0.1\). In percent form, \(P_1 = 10.0\%\).

Step3: Probability that student has competed in all three categories

The number of students who competed in all three categories is \(3\). The probability \(P_2=\frac{3}{50}=0.06\). In percent form, \(P_2 = 6.0\%\).

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

The number of students who competed in Smooth only \(=15\) and Standard only \(=6\). So the number of favorable students \(n = 15+6=21\). The probability \(P_3=\frac{21}{50}=0.42\). In percent form, \(P_3 = 42.0\%\).

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

The number of students who competed in \(R\cap T\) but not in \(S\) is \(3\). The probability \(P_4=\frac{3}{50}=0.06\). In percent form, \(P_4 = 6.0\%\).

Step6: Probability that student has competed in Rhythm

The number of students who competed in Rhythm is \(7 + 8+3=18\). The probability \(P_5=\frac{18}{50}=0.36\). In percent form, \(P_5 = 36.0\%\).

Answer:

a) \(10.0\%\)
b) \(6.0\%\)
c) \(42.0\%\)
d) \(6.0\%\)
e) \(36.0\%\)