There are 300 students at a music college. All students play exactly one of the guitar, the piano or the flute. The numbers of male and female students that play each of the instruments are given in the following table. Guitar Piano Flute --- --- --- --- Female students 62 35 43 Male students 78 40 42 (i) Find the probability that a randomly chosen student at the college is a male who does not play the piano. (ii) Determine whether the events ‘a randomly chosen student is…

(i) 0.4 · (ii) The events are independent

Mathematics 9709/63 — October/November 2019

Cambridge AS Level · Probability & Statistics 1 · worked solutions for every part, with the mark scheme

questions

marks

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Topics Probability · Permutations and Combinations · The Normal Distribution · Discrete Random Variables · Representation of Data

Q1ProbabilityFree sample

There are 300 students at a music college. All students play exactly one of the guitar, the piano or the flute. The numbers of male and female students that play each of the instruments are given in the following table.

	Guitar	Piano	Flute
Female students	62	35	43
Male students	78	40	42

(i)

Find the probability that a randomly chosen student at the college is a male who does not play the piano.

DifficultyEasy

Worked solution

Approach

Identify the number of male students who do not play the piano, then divide by the total number of students.

Working

From the table:

Total students: $300$ .
Male students who do not play the piano: males playing guitar + males playing flute $= 78 + 42 = 120$ .

Therefore,

P(\text{male and not piano}) = \frac{120}{300} = 0.4

Answer

0.4

Final answer

0.4

Detailed explanation

Walkthrough

The table separates students by gender and instrument. 'Male who does not play the piano' refers to male students in the Guitar and Flute columns, since Guitar and Flute are the instruments that are not piano. Add those counts: $78 + 42 = 120$ . A probability is the number of favourable outcomes divided by the total number of equally likely outcomes, so divide $120$ by the total number of students, $300$ . This gives $0.4$ .

Key Takeaways

The probability of an event can be found by counting favourable outcomes from a two-way table and dividing by the total number of outcomes. Recognising which table cells match the description is the key skill.

Common Mistakes

Counting all $160$ male students instead of only those not playing the piano.
Counting only one instrument's male students, such as only male guitarists.
Using $160$ as the denominator instead of $300$ .

Things to Be Careful About

'Does not play the piano' includes both guitar and flute players. The denominator is the total number of students, $300$ , not the number of male students.

Techniques used

read favourable outcomes from a two-way tabledivide the favourable count by the total number of students

(ii)

Determine whether the events ‘a randomly chosen student is male’ and ‘a randomly chosen student does not play the piano’ are independent, justifying your answer.

DifficultyMedium-Easy

Worked solution

Approach

Let $M$ be the event 'the student is male' and $N$ be the event 'the student does not play the piano'. To test independence, calculate $P(M)$ , $P(N)$ and $P(M \cap N)$ , then check whether

P(M \cap N) = P(M)P(N).

Working

From the totals:

Total number of students: $300$ .
Number of male students: $78 + 40 + 42 = 160$ .
Number of students not playing the piano: $(62 + 78) + (43 + 42) = 225$ .
Number of male students not playing the piano: $78 + 42 = 120$ .

Hence

P(M) = \frac{160}{300} = \frac{8}{15}

P(N) = \frac{225}{300} = \frac{3}{4}

P(M \cap N) = \frac{120}{300} = \frac{2}{5}

Now calculate the product:

P(M)P(N) = \frac{8}{15} \times \frac{3}{4} = \frac{24}{60} = \frac{2}{5}

Since

P(M)P(N) = \frac{2}{5} = P(M \cap N),

the two events are independent.

Answer

The events are independent.

Final answer

The events are independent

Detailed explanation

Walkthrough

In part (ii), the goal is to test whether being male and not playing the piano are independent. Start by naming the two events. Independence means the probability of both events occurring equals the product of their individual probabilities. First extract from the table the marginal totals: number of males $160$ , number not playing piano $225$ , and the intersection count $120$ , all out of $300$ . Turn each into a probability. Then multiply $P(M)$ and $P(N)$ and compare the result with $P(M \cap N)$ . Since both equal $\frac{2}{5}$ , the events are independent. If the two values had been different, the events would not be independent.

Key Takeaways

Two events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$ .
A two-way table provides all marginal and joint counts needed to test independence.
Independence is a statement about probabilities, not about whether the events can occur together.

Common Mistakes

Forgetting to include non-piano females when finding $P(N)$ .
Comparing $P(M \cap N)$ with only $P(M)$ or only $P(N)$ .
Writing a conclusion without showing the numerical comparison; the mark scheme requires both a named product and a numerical comparison with a correct conclusion.
Calling the events 'mutually exclusive' instead of 'independent'.

Things to Be Careful About

All probabilities should have denominator $300$ before simplifying.
The events must be named clearly when forming the product $P(M)P(N)$ .
"Does not play the piano" means Guitar or Flute, not piano players.
In this case the equality is exact; do not round in a way that hides the exact $\frac{2}{5}$ comparison.

Techniques used

extract marginal and joint probabilities from a two-way tabletest independence by comparing the joint probability with the product of marginal probabilitiesdraw a conclusion from the numerical comparison

The rest of this paper

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