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

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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 one divided by two, and the probability of scoring two heads is

 0.5 × 0.5 = 0.25,

or

equation left hand side one divided by two multiplication one divided by two equals right hand side one divided by four full stop

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, one divided by four, 0.25 or 25%.

Activity 6 Combining probability

Timing: Allow approximately 20 minutes.

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 one divided by six. For two dice, applying the formula gives an overall probability of

    equation sequence one divided by six multiplication one divided by six equals one divided by 36 equals 0.028 left parenthesis to two s full stop f full stop right parenthesis full stop

  • b.Similarly, for scoring six 6s, the overall probability is

    multiline equation line 1 equation left hand side one divided by six multiplication one divided by six multiplication one divided by six multiplication one divided by six multiplication one divided by six multiplication one divided by six equals right hand side open one divided by six close super six line 2 equation left hand side equals right hand side one divided by 46 postfix times 656 line 3 equation left hand side equals right hand side 0.000 postfix times 021 left parenthesis to two s full stop f full stop right parenthesis full stop

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.