Example 5 Combining two independent probabilities
When tossing a coin twice, what is the probability of getting heads both times?
Solution
Using the formula, the probability of each toss scoring a head is 1 in 2, 0.5 or , and the probability of scoring two heads is
0.5 × 0.5 = 0.25,
or
This can be checked by working from first principles. Tossing two coins (or one coin twice) gives the following four possible outcomes:
- heads, heads
- tails, tails
- heads, tails
- tails, heads.
From this, it can be seen that the probability of getting two heads is 1 in 4, , 0.25 or 25%.
Activity 6 Combining probability
Calculate the probability of the following:
- a.scoring two 6s by rolling two dice
- b.scoring six 6s by rolling six dice.
Express your answers as both a fraction and a decimal (to 2 s.f.).
Answer
- a.The probability of scoring a 6 with one roll of a die is . For two dice, applying the formula gives an overall probability of
- b.Similarly, for scoring six 6s, the overall probability is
Note that if you were to carry out these calculations using the rounded decimal value of 0.17 for the probability of a 6 (as calculated in Activity 5), you would have got the values 0.029 and 0.000 024, respectively. As this demonstrates, using rounded figures in calculations can have a significant effect.