18M.1.SL.TZ2.S_8

pestleMathematicsAISLPaper 118M· sl-4-5-probability-concepts-expected-numberssource ↗

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]

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

[2]

Find the probability that Pablo is late for work.

[3]

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

[3]

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

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

A1A1A1 N3

Note: Award A1 for each bold fraction.

[3 marks]

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]

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]

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]

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]

Examiners’ report

[N/A]