Combined Probabilities (Cambridge International IGCSE Mathematics)

Combined Probability

How do I calculate combined probabilities?

• You can calculate probabilities of one event after another without needing tree diagrams

  • These are called combined (or successive) probabilities

• There are two rules to learn

  • And means multiply and or means add
  • P(A and B) = P(A) x P(B)
  • P(AA or BB) = P(AA) + P(BB)

• Try to rephrase each question using and / or

  • For example, when flipping a coin twice:

    • P(two heads) = P(head and head)
    • P(both the same) = P(head and head or tail and tail) = P(HH) + P(TT)

• Remember that P(not A) = 1 - P(A)

What does independent mean?

• Independent events are events that do not affect each other

  • e.g. the probability of rolling a 6 on a fair dice and the probability of getting a head when flipping a coin

• Be careful: questions 'without replacement' are not independent

  • e.g. the probability of taking a red card out of a pack, not replacing it, then finding the probability of taking a second red card out of the same pack

    • The first event affected the number of cards left for the second event

Worked Example A box contains 3 blue counters and 8 red counters. A counter is taken at random and its colour is noted. The counter is put back into the box. A second counter is then taken at random, and its colour is noted.

Work out the probability that

(a) both counters are red,

Answer:

P(both red) = P(red and red) This is P(red) × P(red) using the 'and rule'

$\begin{array}{rcl} \frac{8}{11} \times \frac{8}{11} \end{array}$

$\begin{gathered} \frac{64}{121} \end{gathered}$

(b) the two counters are of different colours.

Answer:

P(different colours) = P(blue and red or red and blue) This is P(B and R) + P(R and B) using the 'or rule' This is P(B) × P(R) + P(R) × P(B) using the 'and rule' twice

$\begin{gathered} \frac{8}{11} \times \frac{3}{11} + \frac{3}{11} \times \frac{8}{11} \\ = \frac{24}{121} + \frac{24}{121} \end{gathered}$

$\begin{gathered} \frac{48}{121} \end{gathered}$