IB Math Analysis & Approaches Question bank-Topic: SL 4.6- Mutually exclusive events SL Paper 1

Question

Let A and B be independent events, where \({\text{P}}(A) = 0.6\) and \({\text{P}}(B) = x\) .

Write down an expression for \({\text{P}}(A \cap B)\) .

[1]
a.

Given that \({\text{P}}(A \cup B) = 0.8\) ,

(i)     find x ;

(ii)    find \({\text{P}}(A \cap B)\) .

[4]
b(i) and (ii).

Hence, explain why A and B are not mutually exclusive.

[1]
c.
Answer/Explanation

Markscheme

\({\text{P}}(A \cap B) = {\text{P}}(A) \times {\text{P}}(B)( = 0.6x)\)    A1     N1

[1 mark]

a.

(i) evidence of using \({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) – {\text{P}}(A){\text{P}}(B)\)     (M1)

correct substitution     A1

e.g. \(0.8 = 0.6 + x – 0.6x\) , \(0.2 = 0.4x\)

\(x = 0.5\)    A1     N2

(ii) \({\text{P}}(A \cap B) = 0.3\)     A1     N1

[4 marks]

b(i) and (ii).

valid reason, with reference to \({\text{P(}}A \cap B)\)     R1     N1

e.g. \({\text{P(}}A \cap B) \ne 0\)

[1 mark]

c.

Question

In a class of 100 boys, 55 boys play football and 75 boys play rugby. Each boy must play at least one sport from football and rugby.

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

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

[3]
a.

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.

[4]
b.

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.

[2]
c.

Show that A and B are not independent.

[3]
d.
Answer/Explanation

Markscheme

(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]

a.

(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]  

b.

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]

c.

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]

d.

Question

In a group of 20 girls, 13 take history and 8 take economics. Three girls take both history and economics, as shown in the following Venn diagram. The values \(p\) and \(q\) represent numbers of girls.

M17/5/MATME/SP1/ENG/TZ1/01

Find the value of \(p\);

[2]
a.i.

Find the value of \(q\).

[2]
a.ii.

A girl is selected at random. Find the probability that she takes economics but not history.

[2]
b.
Answer/Explanation

Markscheme

valid approach     (M1)

eg\(\,\,\,\,\,\)\(p + 3 = 13,{\text{ }}13 – 3\)

\(p = 10\)     A1     N2

[2 marks]

a.i.

valid approach     (M1)

eg\(\,\,\,\,\,\)\(p + 3 + 5 + q = 20,{\text{ }}10 – 10 – 8\)

\(q = 2\)     A1     N2

[2 marks]

a.ii.

valid approach     (M1)

eg\(\,\,\,\,\,\)\(20 – p – q – 3,{\text{ }}1 – \frac{{15}}{{20}},{\text{ }}n(E \cap H’) = 5\)

\(\frac{5}{{20}}\,\,\,\left( {\frac{1}{4}} \right)\)     A1     N2

[2 marks]

b.

Question

Pablo drives to work. The probability that he leaves home before 07:00 is \(\frac{3}{4}\).

If he leaves home before 07:00 the probability he will be late for work is \(\frac{1}{8}\).

If he leaves home at 07:00 or later the probability he will be late for work is \(\frac{5}{8}\).

Copy and complete the following tree diagram.

[3]
a.

Find the probability that Pablo leaves home before 07:00 and is late for work.

[2]
b.

Find the probability that Pablo is late for work.

[3]
c.

Given that Pablo is late for work, find the probability that he left home before 07:00.

[3]
d.

Two days next week Pablo will drive to work. Find the probability that he will be late at least once.

[3]
e.
Answer/Explanation

Markscheme

A1A1A1 N3

Note: Award A1 for each bold fraction.

[3 marks]

a.

multiplying along correct branches      (A1)
eg  \(\frac{3}{4} \times \frac{1}{8}\)

P(leaves before 07:00 ∩ late) = \(\frac{3}{32}\)    A1 N2

[2 marks]

b.

multiplying along other “late” branch      (M1)
eg  \(\frac{1}{4} \times \frac{5}{8}\)

adding probabilities of two mutually exclusive late paths      (A1)
eg  \(\left( {\frac{3}{4} \times \frac{1}{8}} \right) + \left( {\frac{1}{4} \times \frac{5}{8}} \right),\,\,\frac{3}{{32}} + \frac{5}{{32}}\)

\({\text{P}}\left( L \right) = \frac{8}{{32}}\,\,\left( { = \frac{1}{4}} \right)\)    A1 N2

[3 marks]

c.

recognizing conditional probability (seen anywhere)      (M1)
eg  \({\text{P}}\left( {A|B} \right),\,\,{\text{P}}\left( {{\text{before 7}}|{\text{late}}} \right)\)

correct substitution of their values into formula      (A1)
eg \(\frac{{\frac{3}{{32}}}}{{\frac{1}{4}}}\)

\({\text{P}}\left( {{\text{left before 07:00}}|{\text{late}}} \right) = \frac{3}{8}\)    A1 N2

[3 marks]

d.

valid approach      (M1)
eg  1 − P(not late twice), P(late once) + P(late twice)

correct working      (A1)
eg  \(1 – \left( {\frac{3}{4} \times \frac{3}{4}} \right),\,\,2 \times \frac{1}{4} \times \frac{3}{4} + \frac{1}{4} \times \frac{1}{4}\)

\(\frac{7}{{16}}\)    A1 N2

[3 marks]

e.
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