In a group of students, the number of males and females studying Spanish, French or Music are given in a table below Spanish French Music Males 23 12 26 Females 12 15 09 i) Find the probability that a randomly chosen student a) studies music or is female c) is male, given that they do not study Spanish ii) Determine whether the events ‘a randomly chosen student is a male’ and ‘a randomly chosen student is studying Music’ are independent, justifying your answer. iii) Two students are randomly selected, without replacement, what is the probability that the first student studies Spanish and the second student studies French.

i)                  Find the probability that a randomly chosen student

a)    studies music or is female     

we define P( studies music or female) = P( studies music) + P(female) - P(studies music and female)

P( studies music) = (total of those who study music) /( total number of students)                                         = (26+9) / ( 23+12+26+12+15+9)

            = ( 35/97 ) =0.3608

P(female) = (total number of females)/(total number of students)

= (12+15+9) / (23+12+26+12+15+9)

= ( 36/97 ) = 0.3711

P(studies music and female) =(females studying music)/(total number of students)

= (9/97 ) = 0.09278

Thus P( studies music or female) = P( studies music) + P(female) - P(studies music and female)

= ( 35/97) + (36/97) - (9/97) = ( 62/97 ) = 0.63918

c)    is male, given that they do not study Spanish  

we define

P(  is male, given that they do not study Spanish) = P( they do not study Spanish) / P(male)

P( they do not study Spanish) = (students not studying spanish )/(total number of students)                            = ( 12+26 ) /(97) = 38/97

P(male)= ( total number of males)/(total number of students)

= ( 61/97 )

Thus  P(  is male, given that they do not study Spanish) = P( they do not study Spanish) / P(male)

= (38/97) / (61/97)

=( 38/61 ) = 0.62295

ii)                Determine whether the events ‘a randomly chosen student is a male’ and ‘a randomly chosen student is studying Music’ are independent, justifying your answer. 

we define that for two events to be independent, then it follows that P( A and B) = P(A) * P(B)

now let Male = M and Music = U, we define the relation below

P( M and U) = P(M) * P(U)

We find

P( M and U) =(males who study music)/(total number of students) = 26/97

P(M) =(total number of males)/(total number of students) = 61/97

P(U) = ( students who take music)/ (total number of students)= 35/97

Now P(M) * P(U) = (61/97) * (35/97) = 0.22691

P( M and U) = (26/97) = 0.268041

Thus P( M and U)is not equal to P(M) * P(U), hence we conclude that M and U are not independent events.

iii)              Two students are randomly selected, without replacement, what is the probability that the first student studies Spanish and the second student studies French.      

we define the total outcome of the experiment as ( 97 * 96) =9312

Define event B = the first student studies spanish and the second student studies french                         

  = ( 35 * 27 ) = 945

Thus P(B) = ( 945 ) / ( 9312 ) = 0.101481959 which is the required probability.