Question

Question 1 (30 marks)

(a) 500 adults were asked whether or not they will get married in the next 3 years. Their responses are recorded in the table.

Would marry in the next 3 years

	Yes(Y)	No (N)
Male (M)	150	150
Female (F)	100	100

If one adult is selected at random from these 500 persons, find the probability that

(i) the response is 'Yes'; [2 marks]

(ii) the response is 'Yes' given female; [2 marks]

(iii) the response is 'No' and a male; [2 marks]

(iv) the response is 'Yes' or female. [2 marks]

Are the events 'Male' and 'Yes' independent? Explain why or why not. [2 marks]

Are the events 'Female' and 'No' mutually exclusive? Explain why or why not. [2 marks

(b) The probability that a student selected at random is a male is 0.48, and a student likes statistics is 0.35. The probability of selected at random a female who likes statistics is 0.25.

(i) Construct a contingency table based on the information given above. [2 marks]

(ii) What is the probability that a student selected at random is a male and likes statistics; [2 marks]

(iii) What is the probability that a student selected at random is a female or likes statistics. [2 marks]

(c) The school finds that 30% of the students are low achievers. The Headmaster is instructed to devise a revision module that will improve the situation. One hundred of the students are chosen at random and are given the revision module. Out of these, 90% of the high achievers pass the test and 20% of the low achievers also pass.

Based on these results, if a student chosen at random, takes the module, and passes it, what is the probability that he or she is a high achiever? If the student fails the module, what is the probability that he or she is a high achiever? [12 marks]

Answer 1

The contingency table for the given question is as follows.

	Response
Gender	Yes(Y)	No (N)	Total
Male (M)	150	150	300
Female (F)	100	100	200
Total	250	250	500

Now probability of occurrence of an event is given by the ratio of the total favorable instances/conditions /cases{F} of the given event to the total sum of all instances/conditions /cases events.{T}

Mathematically if A is an event then

P(A) ={F}/{T}

1. We need to find the probability of response of YES from total set of 500 adults.

Clearly if we refer to the table above we can see that total of 250 adults said yes. Thus this will be our favorable cases.

Now the probability of response being yes = Total adults saying yes/total adults

Or, P(yes) =250/500

Or, P(Yes)=1/2

=0.50 or 50 %

ii). We need to find the probability of response of YES from the set of females which is equal to 200.

Clearly if we refer to the table above we can see that total of 100 adult females said yes. Thus this will be our favorable cases.

Now the probability of response being yes = Total female adults saying yes/total female adults

Or, P(yes-female) =100/200

Or, P(Yes-Female)=1/2

=0.50 or 50 %

iii). We need to find the probability of response of NO from the set of males which is equal to 300.

Clearly if we refer to the table above we can see that total of 150 adult males said NO. Thus this will be our favorable cases.

Now the probability of response being NO given it is from male = Total male adults saying NO /total male adults

Or, P(NO-male) =150/300

Or, P(No-Male)=1/2

=0.50 or 50 %

iv). Here we need to find the probability of response being YES or being Female from the entire set.

Here we will have to think !

Firstly we will find the probability of YES

Then we will find the probability of being a female

Now since it is asking us to only consider cases where it is either YES or it is female ,we have to remove all the cases which occur due to they being Yes and female.

Thus we will have

Required probability (P) =P(YES)+P(Female)-P(Yes-Female)

We know all the values in RHS thus keeping values we get

P =P(YES)+P(female)-P(Yes-Female)

Or P =250/500+200/500-100/500

Or, P =1/2+2/5-1/5

Or, P =3.5/5

Or, P =0.70 ~70 %

Thus the probability of yes or female is 70%

v). To check if two events are independent or not, we need to check if the probability that one event occurs is in no way affected by the probability of the other event occurring.

mathematically

for two events A and B

if  P ( A ∩ B ) = P ( A ) P ( B )

lets check it

P(Males) =300/500=0.60

P(yes) =250/500 =0.50

P(Male and Yes) =150/500 =0.30

but P(Male)*P(yes) =0.60*0.50 =0.30

Thus P(Male and Yes) =P(male)*P(yes) proved.

Hence we say that both events(YES,MALE) are independent of each other.

vi). To check if two events re mutual exclusive or not, we need to check if they cannot both occur at the same time (be true).

Mathematically if there be two events A and B

then

P ( A ∩ B ) = P ( A ) P ( B )

Now P(male)=300/500 =0.60

P(yes) =250/500 =0.50

P(yes-Male) =150/500 =0.30

But P(male)*P(yes) =0.60*0.50 =0.30

Thus  P ( male ∩ yes ) = P ( male )* P ( yes)

Hence we say that the two events are independent of each other.

vi). To check if two events re mutual exclusive or not, we need to check if they cannot both occur at the same time (be true).

Clearly if we refer to the table above we can see that total of 100 adult females said NO. Hence the two events are NOT mutually exclusive.