ProbabilityIB MYP Maths Standard: Topic test

1. 1
   A box at a charity raffle in Mumbai holds $20$ tickets numbered $1$ to $20$. One ticket is chosen at random.

   (a)

   What is the probability that the ticket is a multiple of $5$?

   [1 mark]

   • A$\frac{1}{4}$
   • B$\frac{1}{20}$
   • C$\frac{1}{5}$
   • D$\frac{4}{5}$

   (b)

   What is the probability that the ticket is not a prime number?

   [1 mark]

   • A$\frac{3}{5}$
   • B$\frac{2}{5}$
   • C$\frac{11}{20}$
   • D$\frac{3}{4}$

   (c)

   Are the events “the ticket is even” and “the ticket is a multiple of $5$” mutually exclusive? Give a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A bus company in Bogota records that on $120$ of the last $150$ mornings the $7{:}30$ bus arrived on time.

   (a)

   Use the records to estimate the probability that the bus arrives on time.

   [1 mark]

   • A$\frac{1}{5}$
   • B$\frac{4}{5}$
   • C$\frac{150}{120}$
   • D$\frac{120}{270}$

   (b)

   Based on these records, how many on-time arrivals are expected in the next $200$ mornings?

   [1 mark]

   • A$40$
   • B$120$
   • C$250$
   • D$160$

   (c)

   Give one way to improve the reliability of the estimate and explain why it helps.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The universal set $\xi$ is the set of integers from $1$ to $20$. Set $A$ is the set of multiples of $3$ and set $B$ is the set of multiples of $4$.

   (a)

   (i) Write down $A\cap B$. (ii) List the elements of $A\cup B$. (iii) Find $n\big((A\cup B)'\big)$.

   [3 marks]

   (b)

   A number is chosen at random from $\xi$. Find (i) $P(A\cup B)$ and (ii) the probability that the number is in $B$ but not in $A$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school in Sydney has $50$ students. Students may belong to the chess club ($C$), the drama club ($D$) and the robotics club ($R$). There are $16$ students in chess, $17$ in drama and $18$ in robotics. Exactly $2$ students are in all three clubs, $4$ are in chess and drama only, $3$ are in chess and robotics only, and $5$ are in drama and robotics only.

   (a)

   (i) Find the number of students in chess only and the number in none of the three clubs. (ii) A student is chosen at random. Find the probability that the student belongs to exactly two of the clubs.

   [6 marks]

   (b)

   Two students are chosen at random, one after the other, without replacement. Use $15$ as the number of students in no club. (i) Find the probability that both students are in no club. (ii) Find the probability that exactly one of the two students is in no club. (iii) This selection is repeated $245$ times, with both students returned each time. Estimate how many times both students are in no club.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A driver on a road in Singapore meets two traffic lights. The probability that the driver does not have to stop at the first light is $0.6$. The probability of not having to stop at the second light is $0.5$, independent of the first.

   (a)

   What is the probability that the driver does not have to stop at either light?

   [1 mark]

   • A$0.3$
   • B$1.1$
   • C$0.1$
   • D$0.55$

   (b)

   What is the probability that the driver has to stop at exactly one of the two lights?

   [1 mark]

   • A$0.3$
   • B$0.2$
   • C$0.5$
   • D$0.9$

   (c)

   Find the probability that the driver has to stop at least once.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A cafe in Montreal has $36$ customers at lunchtime. Of these, $20$ ordered coffee, $15$ ordered tea and $6$ ordered both.

   (a)

   How many customers ordered coffee only?

   [1 mark]

   • A$20$
   • B$9$
   • C$6$
   • D$14$

   (b)

   A customer is chosen at random. What is the probability that they ordered neither coffee nor tea?

   [1 mark]

   • A$\frac{29}{36}$
   • B$\frac{7}{36}$
   • C$\frac{7}{29}$
   • D$\frac{14}{36}$

   (c)

   A customer is chosen at random. Find the probability that they ordered exactly one of the two drinks.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A game developer in Warsaw tests a card-dealing program. In $500$ trials it deals one card from a shuffled standard pack of $52$ cards (which contains $4$ aces). After each trial the card is returned and the pack is reshuffled. An ace was dealt $52$ times.

   (a)

   (i) Calculate the relative frequency of an ace in the trials. (ii) Calculate the theoretical probability of dealing an ace.

   [3 marks]

   (b)

   Calculate the number of aces expected in $500$ trials and compare it with the number dealt. Does the test prove that the program is unfair? Justify your answer.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A farmer in Kenya plants $2$ seeds in each of $250$ holes. Each seed germinates with probability $0.8$, independently of the other seeds.

   (a)

   Use a tree diagram or the rules of probability for one hole to find (i) the probability that both seeds germinate, (ii) the probability that exactly one seed germinates and (iii) the probability that at least one seed germinates.

   [6 marks]

   (b)

   Using $0.96$ as the probability that a hole has at least one plant, calculate how many of the $250$ holes are expected to have a plant. The farmer considers planting a third seed in every hole. Each extra seed costs $0.05$ dollars and each hole that grows a plant earns $2.00$ dollars. Evaluate whether the third seed is worthwhile, showing your calculations.

   [6 marks]

   Total for question 8: 12 marks

End of questions