The probability that the student competed in Rhythm given they had competed in Smooth or Standard: P(Rhythm |Smooth or Standard) or P ( A/( B or C)) is 0.6613 = 66.13%.
We have, A survey was conducted at a local ballroom dance studio asking students with three dance categories. The above Venn diagram shows the number of students in these three categories of dance.
Total number of students in all dance categories n(A or B or C) = 5 + 6 + 8 + 12 + 14 + 15 + 10 + 4
= 74
let us consider the events
A : students in Rhythm dance category
B : students in Standard dance category
C : students in Smooth dance category
Using the conditional probability,
Probability that student has competed in Rhythm and they have competed in Smooth, P(Rhythm/ smooth) = P(A/C) = P(A and C)/P(C)
= (12+14)/74 = 26/74 = 0.703
but we have to calculate probability that the student competed in Rhythm given they had competed in Smooth or Standard, P(Rhythm/Smooth or Standard) = P(Rhythm and (Smooth or Standard))/P(Smooth or Standard)
i.e., P( A/(B or C) )= P(A and (B or C))/P(B or C)
n(B or C) = n(B) + n(C) - n( B and C)
=> n(B or C) = 14 + 5 + 10 + 15 + 12 + 14 + 5 + 6 - 5 - 14 = 62
Now, P(B or C) =n(B or C)/n(A or B or C)
=> P(B or C) = 62/74 = 0.703 = 70.3%
P(A and (B or C)) = n(A and (B or C))/n(A or B or C)
n(A and (B or C)) = n(A) + n(B or C) - n(A or B or C)
= 12 + 14 + 15 + 8 + 62 - 70 +4 = 41
P(A and (B or C)) = 37/74
So, P( A/(B or C) )= P(A and (B or C))/P(B or C)
=> P( A/(B or C) )= 41/74/62/74 = 41/62
= 0.6613 = 66.13%
Hence, the required probability is 66.13%.
To learn more about conditional probability, refer:
https://brainly.com/question/10567654
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Complete 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 or Standard: P(Rhythm |Smooth or Standard)
