(i)     Find the number of boys who play both sports.

(ii)    Write down the number of boys who play only rugby.

One boy is selected at random.

(i)     Find the probability that he plays only one sport.

(ii)    Given that the boy selected plays only one sport, find the probability that he plays rugby.

Let A be the event that a boy plays football and B be the event that a boy plays rugby.

Explain why A and B are not mutually exclusive.

Show that A and B are not independent.

(i) evidence of substituting into \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)     (M1)

e.g. \(75 + 55 - 100\) , Venn diagram

30     A1     N2 

(ii) 45     A1     N1

[3 marks]

(i) METHOD 1

evidence of using complement, Venn diagram     (M1)

e.g. \(1 - p\) , \(100 - 30\)

\(\frac{{70}}{{100}}\) \(\left( { = \frac{7}{{10}}} \right)\)     A1     N2

METHOD 2

attempt to find P(only one sport) , Venn diagram     (M1) 

e.g. \(\frac{{25}}{{100}} + \frac{{45}}{{100}}\)

\(\frac{{70}}{{100}}\) \(\left( { = \frac{7}{{10}}} \right)\)     A1     N2

(ii) \(\frac{{45}}{{70}}\) \(\left( { = \frac{9}{{14}}} \right)\)     A2     N2

[4 marks]  

valid reason in words or symbols     (R1)

e.g. \({\rm{P}}(A \cap B) = 0\) if mutually exclusive, \({\rm{P}}(A \cap B) \ne 0\) if not mutually exclusive

correct statement in words or symbols     A1     N2

e.g. \({\rm{P}}(A \cap B) = 0.3\) , \({\rm{P}}(A \cup B) \ne {\rm{P}}(A) + {\rm{P}}(B)\) , \({\rm{P}}(A) + {\rm{P}}(B) > 1\) , some students play both sports, sets intersect

[2 marks]

valid reason for independence     (R1)

e.g. \({\rm{P}}(A \cap B) = {\rm{P}}(A) \times {\rm{P}}(B)\) , \({\rm{P}}(B|A) = {\rm{P}}(B)\)

correct substitution     A1A1     N3

e.g. \(\frac{{30}}{{100}} \ne \frac{{75}}{{100}} \times \frac{{55}}{{100}}\) , \(\frac{{30}}{{55}} \ne \frac{{75}}{{100}}\)

[3 marks]

Overall, this question was very well done. There were some problems with the calculation of conditional probability, where a considerable amount of candidates tried to use a formula instead of using its concept and analysing the problem. It is the kind of question where it can be seen if the concept is not clear to candidates.

Overall, this question was very well done. There were some problems with the calculation of conditional probability, where a considerable amount of candidates tried to use a formula instead of using its concept and analysing the problem. It is the kind of question where it can be seen if the concept is not clear to candidates.

In part (c), candidates were generally able to explain in words why events were mutually exclusive, though many gave the wrong values for P(A) and P(B).

There was a great amount of confusion between the concepts of independent and mutually exclusive events. In part (d), the explanations often referred to mutually exclusive events.

It was evident that candidates need more practice with questions like (c) and (d).

Some students equated probabilities and number of elements, giving probabilities greater than 1.