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

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Assessing risk in engineering, work and life

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

    multiline equation line 1 equation left hand side one divided by six multiplication one divided by six equals right hand side one divided by 36 full stop

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

    multiline equation line 1 one divided by six plus one divided by six minus one divided by 36 equation left hand side equals right hand side six divided by 36 plus six divided by 36 minus one divided by 36 line 2 equation left hand side equals right hand side 11 divided by 36 line 3 equals 0.31 left parenthesis to two s full stop f full stop right parenthesis full stop
  • 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

    multiline equation line 1 equation left hand side five divided by six multiplication five divided by six multiplication five divided by six equals right hand side 125 divided by 216 full stop

    So the probability of scoring at least one 2 is

    multiline equation line 1 equation left hand side one minus 125 divided by 216 equals right hand side 216 divided by 216 minus 125 divided by 216 line 2 equation left hand side equals right hand side 91 divided by 216 line 3 equals 0.42 left parenthesis to two s full stop f full stop right parenthesis full stop

Activity 8 Probabilities

Timing: Allow approximately 25 minutes.

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

multiline equation line 1 equation left hand side one divided by two multiplication one divided by six equals right hand side one divided by 12 full stop

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

multiline equation line 1 equation left hand side one divided by two plus one divided by six minus one divided by 12 equals right hand side six divided by 12 plus two divided by 12 minus one divided by 12 line 2 equation left hand side equals right hand side seven divided by 12 line 3 equals 0.58 left parenthesis to two s full stop f full stop right parenthesis full stop

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:

    multiline equation line 1 equation left hand side two multiplication 10 super negative five plus two multiplication 10 super negative three minus four multiplication 10 super negative eight equals right hand side open 2000 plus 200 postfix times 000 minus four close multiplication 10 super negative eight line 2 equation left hand side equals right hand side 201 postfix times 996 multiplication 10 super negative eight line 3 equation left hand side equals right hand side 2.019 postfix times 96 multiplication 10 super negative three line 4 equation left hand side equals right hand side 2.0 multiplication 10 super negative three left parenthesis to two s full stop f full stop right parenthesis full stop

    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

    multiline equation line 1 equation left hand side one minus cap p of defect equals right hand side one minus 2.01996 multiplication 10 super negative three line 2 equals 0.998 left parenthesis to three s full stop f full stop right parenthesis full stop

T193_1

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