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. use the venn diagram to determine the following probabilities. write your answers in percent form, rounded to the nearest tenth. if a student is chosen at random, what is the probability that the student competed in rhythm given they had competed in smooth. p(rhythm | smooth) = 15 % if a student is chosen at random, what is the probability that the student competed in rhythm given they had competed in standard. p(rhythm | standard) = 11 % if a student is chosen at random, what is the probability that the student competed in standard given they had competed in rhythm. p(standard | rhythm) = % if a student is chosen at random, what is the probability that the student competed in rhythm given they had competed in smooth or standard. p(rhythm | smooth or standard) = %

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In terms of the Venn - diagram, if we want to find $P(A|B)$, we use the number of elements in the intersection of $A$ and $B$ divided by the number of elements in $B$.

Step2: Calculate $P(\text{Standard}|\text{Rhythm})$

The number of students who competed in both Standard and Rhythm is $12 + 9=21$. The number of students who competed in Rhythm is $10 + 15+9 + 12=46$. Then $P(\text{Standard}|\text{Rhythm})=\frac{21}{46}\approx0.457$

Step3: Convert to percentage

To convert the decimal to a percentage, we multiply by 100. So $P(\text{Standard}|\text{Rhythm}) = 45.7\%$

Step4: Calculate $P(\text{Rhythm}|\text{Smooth or Standard})$

The number of students who competed in Smooth or Standard is $15 + 15+5 + 9+12 + 11=67$. The number of students who competed in both Rhythm and (Smooth or Standard) is $15 + 9+12=36$. Then $P(\text{Rhythm}|\text{Smooth or Standard})=\frac{36}{67}\approx0.537$

Step5: Convert to percentage

Multiply the decimal by 100. So $P(\text{Rhythm}|\text{Smooth or Standard}) = 53.7\%$

Answer:

$P(\text{Standard}|\text{Rhythm}) = 45.7\%$
$P(\text{Rhythm}|\text{Smooth or Standard}) = 53.7\%$