Skip to main content

About this free course

Download this course

Share this free course

Assessing risk in engineering, work and life
Assessing risk in engineering, work and life

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

Combining probabilities

In risk assessments, it is rare to be concerned with the probability that a single event will happen or a single component will fail. As you saw in the Deepwater Horizon case study, several failures led to the incident. Therefore it is often necessary to look at the probability of combined events.

Combining independent probabilities

For any event cap a that has probability cap p of cap a, the probability that cap a does not occur is written as of cap p left parenthesis right parenthesis cap a macron. If the probability is expressed as a fraction or a decimal then cap p times open cap a close postfix plus times of cap p left parenthesis right parenthesis cap a macron equals one. This can be represented using a ‘probability space diagram’, as shown in Figure 15(a). The circle represents the probability that cap a occurs, while the space outside the circle represents the probability that it does not occur.

If two events are independent (the outcome of one does not influence the outcome of the other), the probability of both events occurring is given by

multiline equation row 1 overall probability of two equals probability of multiplication probability of row 2 independent events occurring the first event the second event full stop

For more than two events, the overall probability is given by multiplying the probabilities of each of the events expressed as a fraction or a decimal.

Using standard probability notation,

equation left hand side cap p times open cap a intersection cap b close equals right hand side cap p of cap a multiplication cap p of cap b comma

where the symbol intersection means AND. This expression can be read as ‘the probability of cap a and cap b occurring is equal to the probability of cap a occurring multiplied by the probability of cap b occurring’. Again, this can be represented using a probability space diagram, as shown in Figure 15(b). Here there is one circle representing the probability of cap a occurring and another representing the probability of cap b occurring. The intersection of the two circles (shaded darker blue) represents cap p times open cap a intersection cap b close.

Described image
Figure 15 Probability space diagram for (a) cap p times open cap a close postfix plus times of cap p left parenthesis right parenthesis cap a macron equals one; (b) cap p times open cap a intersection cap b close

Note that if two probabilities are dependent (the outcome of one does influence the outcome of the other), a different method has to be used to combine them, which is beyond the scope of this course. For example, suppose you draw two cards from a single pack of 52 playing cards – what is the probability that they will both be queens? The probability of the first card you draw being a queen is straightforward: there are four queens in a pack of 52 cards, so the probability is 4 in 52, or 1 in 13. However, when you draw the second card, the probability of drawing a queen will either be 3 in 51 (if the first card was a queen) or 4 in 51 (if the first card was not a queen). So the first event is affecting the outcome of the second – the events are dependent.