Copy and complete the following tree diagram.

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

Find the probability that Pablo is late for work.

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

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

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]