Conditional probabilityIB MYP Maths Extended: Subtopic test

1. 1
   A school in Nairobi has 80 students in Years 9 and 10 who were asked whether they play a musical instrument. In Year 9, 18 of 40 students play an instrument and 22 do not. In Year 10, 27 of 40 students play an instrument and 13 do not. One student is chosen at random from the 80.

   (a)

   Given that the student is in Year 10, what is the probability that the student plays an instrument?

   [1 mark]

   • A$\frac{27}{80}$
   • B$\frac{27}{40}$
   • C$\frac{27}{45}$
   • D$\frac{45}{80}$

   (b)

   Given that the student plays an instrument, what is the probability that the student is in Year 9?

   [1 mark]

   • A$\frac25$
   • B$\frac{9}{20}$
   • C$\frac{9}{40}$
   • D$\frac35$

   (c)

   Find the probability that a student who does not play an instrument is in Year 10.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student investigates bags that each contain $r$ red counters and 1 blue counter. She picks two counters without replacement and finds the probability that both are red. For $r=2$ it is $\frac13$, for $r=3$ it is $\frac12$, for $r=4$ it is $\frac35$ and for $r=5$ it is $\frac23$.

   (a)

   What is the probability that both counters are red when $r=9$?

   [1 mark]

   • A$\frac{9}{10}$
   • B$\frac89$
   • C$\frac{10}{11}$
   • D$\frac45$

   (b)

   Which rule gives the probability that both counters are red for every value of $r$ in the data?

   [1 mark]

   • A$\frac{r}{r+1}$
   • B$\frac{r-1}{r}$
   • C$\frac{r-1}{r+1}$
   • D$\frac{1}{r+1}$

   (c)

   Use the pattern to find the value of $r$ for which the probability that both counters are red is $\frac{9}{11}$, and verify your answer by calculating the probability directly.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A basket in a market in Cairo holds 10 mangoes: 6 are ripe and 4 are unripe. Two mangoes are taken at random without replacement.

   (a)

   Find the probability that the first mango is ripe, the probability that the second is unripe given that the first was ripe, and the probability that the first is ripe and the second is unripe.

   [3 marks]

   (b)

   Find the probability that the two mangoes have the same ripeness.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a group of 50 students at an international school in Singapore, 30 study French, 22 study Spanish and 8 study both. A teacher says: 'Students who study French are more likely to study Spanish than students in this group as a whole.'

   (a)

   One student is chosen at random from the group.
   (i) Find the number of students who study neither French nor Spanish.
   (ii) Find the probability that the student studies Spanish, given that they study French.
   (iii) Find the probability that the student studies French, given that they study Spanish.

   [6 marks]

   (b)

   Use probabilities to evaluate the teacher's claim, and suggest one possible reason for your finding.

   [6 marks]

   Total for question 4: 12 marks

End of questions