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

We sum up all the numbers in the Venn - diagram and the number outside the circles. The total number of students $N=11 + 12+5 + 7+9+13+8+6=61$.

Step2: Calculate probability for part a

The number of students who have competed in none of the categories is 6. The probability $P(a)=\frac{6}{61}\approx 0.0984$, and in percent form $P(a)\approx 9.8\%$.

Step3: Calculate probability for part b

The number of students who have competed in all three categories is 9. The probability $P(b)=\frac{9}{61}\approx 0.1475$, and in percent form $P(b)\approx 14.8\%$.

Step4: Calculate probability for part c

The number of students who have competed in Smooth or Standard but not Rhythm is $11 + 7=18$. The probability $P(c)=\frac{18}{61}\approx 0.2951$, and in percent form $P(c)\approx 29.5\%$.

Step5: Calculate probability for part d

The number of students who have competed in Rhythm and Standard but not Smooth is 13. The probability $P(d)=\frac{13}{61}\approx 0.2131$, and in percent form $P(d)\approx 21.3\%$.

Step6: Calculate probability for part e

The number of students who have competed in Rhythm is $12 + 5+9+13 = 39$. The probability $P(e)=\frac{39}{61}\approx 0.6393$, and in percent form $P(e)\approx 63.9\%$.

Answer:

a) $9.8\%$
b) $14.8\%$
c) $29.5\%$
d) $21.3\%$
e) $63.9\%$