a survey was conducted at a local ballroom dance studio asking students if they had ever competed in the…

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. %

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. %

Answer

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%)