19M.2.AHL.TZ2.H_3

pestleMathematicsAIHLPaper 219M· sl-4-5-probability-concepts-expected-numberssource ↗

Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.

Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.

[3]

Calculate the probability that Iqbal passes at least two of the papers he attempts.

[2]

Find the probability that Iqbal passes his third paper, given that he passed only one previous paper.

[3]

Markscheme / solution

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

A1A1A1

Note: Award A1 for each correct column of probabilities.

[3 marks]

probability (at least twice) =

EITHER

$\left( {0.6 \times 0.7 \times 0.8} \right) + \left( {0.6 \times 0.7 \times 0.2} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)$ (M1)

$\left( {0.6 \times 0.7} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)$ (M1)

Note: Award M1 for summing all required probabilities.

THEN

= 0.696 A1

[2 marks]

P(passes third paper given only one paper passed before)

$= \frac{{{\text{P}}\,\left( {{\text{passes third AND only one paper passed before}}} \right)}}{{{\text{P}}\,\left( {{\text{passes once in first two papers}}} \right)}}$ (M1)

$= \frac{{\left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)}}{{\left( {0.6 \times 0.3} \right) + \left( {0.4 \times 0.6} \right)}}$ A1

= 0.657 A1

[3 marks]

Examiners’ report

[N/A]