# Example 6 Calculating the probability of at least one event occurring

What is the probability of each of the following?

- a.Scoring at least one 2 by rolling two dice
- b.Scoring at least one 2 by rolling three dice

**Solution**

The probability of scoring any given number when rolling one die is 1 in 6.

a.The probability of scoring two of any given number by rolling two dice is given by

Using the formula, a probability of at least one 2 is given by

b.For the second case, with three dice, it is necessary to look at the probability of each event not occurring. This is 5 in 6 for each die. So the probability of not scoring any 2s is

So the probability of scoring at least one 2 is

## Activity 8 Probabilities

If a person tosses a coin once and rolls a die once, what is the probability that they get one head and/or one 6? Express your answer as a percentage.

### Answer

The probabilities of scoring a head and rolling a six are 1 in 2 and 1 in 6, respectively, so the probability of both occurring is

Applying the formula, the probability of at least one of the two events occurring is

In other words, at least one of these events would be expected to occur on just over half the occasions. As a percentage, this is a probability of 58%.

A company manufactures bolts. The probability of a bolt having a defective thread is 2 × 10^{−3} (2 in 1000). The probability of a defect in the head is 2 × 10^{−5} (2 in 100 000).

Calculate:

- a.the probability of a bolt having both a defective head and a defective thread
- b.the probability of a bolt having a defective head or a defective thread (remember that when discussing probability, the term ‘or’ means either or both)
- c.the probability that a bolt has no defects.

### Answer

- a.The probability that a bolt has both defects is the product of the two probabilities:
2 × 10

^{−5}× 2 × 10^{−3}= 4 × 10^{−8}This is very small indeed.

- b.To find the probability of a bolt having either (or both) of the defects, you need to use the formula for the probability of at least one event occurring. This is the sum of the two probabilities, minus the probability of both:
This is approximately the same as the probability of a thread defect, which is the dominant defect, so the result shouldn’t surprise you.

- c.In Part (b) you calculated the probability of a defect, cap p of defect . So the probability of no defects is given by