The table shows the percentage of students in each of three

The table shows the percentage of students in each of three grade levels who list soccer as their favorite sport.

Find the probability, rounded to the nearest hundredth, that any student selected from all three grades is a junior who lists soccer as the favorite sport

Solution

let S be the event that a randomly selected student is Sophomore.

J be the event that a randomly selected student is Junior.

T be the event that a randomly selected student is senior.

and let A be the event that a randomly selected student list soccer as their favourite sport.

now given in tha question

P[S]=35%=0.35

P[J]=33%=0.33

P[T]=32%=0.32

P[a student list soccer as his favourite sport given that he is sophomore]=P[A|S]=50%=0.5

P[a student list soccer as his favourite sport given that he is junior]=P[A|J]=45%=0.45

P[a student list soccer as his favourite sport given that he is senior]=P[A|T]=30%=0.30

so

P[a student list soccer as his favourite sport]=P[A]=P[A|S]*P[S]+P[A|J]*P[J]+P[A|T]*P[T] [by theorem of total probability]

                                                                          =(0.5)(0.35)+(0.45)(0.33)+(0.32)(0.3)= 0.4195 or about 42%

so P[A]=42%=0.42

now we need to find

P[any student selected from all three grades is a junior who lists soccer as the favorite sport]

=P[J|A]=P[A|J]*P[J]/P[A]   [by Bayes\' theorem]

           =0.45*0.33/0.42=0.35357143 or about 35% [rounded to nearest hundreth]

so the probability that any student selected from all three grades is a junior who lists soccer as the favorite sport is 35% [answer]