Probability of Combined Events practice questions

Secondary 4 topics from the O-Level Mathematics 4052 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

Practise probability of combined events free

What this topic covers

Worked examples

Example 1

A box of cards numbered 1 to 8 and a fair 10-sided die are used together. Using a possibility diagram or otherwise, find the probability that the DIFFERENCE between the two results, larger minus smaller, is a prime number.
Show the worked solution
  • Total number of outcomes $= 8 \times 10 = 80$
  • The difference runs from $0$ to $9$.
  • Prime differences available: $\{2, 3, 5, 7\}$
  • Favourable outcomes: $(1,3)$,\ $(1,4)$,\ $(1,6)$,\ $(1,8)$,\ $(2,4)$,\ $(2,5)$,\ $(2,7)$,\ $(2,9)$,\ $(3,1)$,\ $(3,5)$,\ $(3,6)$,\ $(3,8)$,\ $(3,10)$,\ $(4,1)$,\ $(4,2)$,\ $(4,6)$,\ $(4,7)$,\ $(4,9)$,\ $(5,2)$,\ $(5,3)$,\ $(5,7)$,\ $(5,8)$,\ $(5,10)$,\ $(6,1)$,\ $(6,3)$,\ $(6,4)$,\ $(6,8)$,\ $(6,9)$,\ $(7,2)$,\ $(7,4)$,\ $(7,5)$,\ $(7,9)$,\ $(7,10)$,\ $(8,1)$,\ $(8,3)$,\ $(8,5)$,\ $(8,6)$,\ $(8,10)$
  • Total favourable outcomes $= 38$
  • $P(\text{difference is prime}) = \frac{19}{40}$

Answer: 19/4019/40

Example 2

A box contains 77 blue pens and 55 red pens. A pen is drawn at random, its colour is noted, and then it is replaced. A second pen is then drawn at random. Find the probability that the second pen drawn is blue.
Show the worked solution
  • Tree diagram: First draw, Blue (probability $\frac{7}{12}$) or Red (probability $\frac{5}{12}$). Since the pen is replaced, the second draw has identical branches.
  • $P(\text{second is blue}) = P(BB) + P(RB)$
  • $= \left(\frac{7}{12} \times \frac{7}{12}\right) + \left(\frac{5}{12} \times \frac{7}{12}\right)$
  • $= \frac{49}{144} + \frac{35}{144}$
  • $= \frac{7}{12}$

Answer: 7/127/12

Example 3

The probability that event AA occurs is 310\frac{3}{10} and the probability that event BB occurs is 15\frac{1}{5}. Events AA and BB are mutually exclusive. Find the probability that AA, BB or both occur.
Show the worked solution
  • Since $A$ and $B$ are mutually exclusive, $P(A \cup B) = P(A) + P(B)$.
  • $P(A \cup B) = \frac{3}{10} + \frac{1}{5}$
  • $= \frac{1}{2}$

Answer: 1/21/2

Questions parents ask

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Which syllabus does this follow?

O-Level Mathematics (4052), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

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