TL Maths

Question of the day 2.09

Tomek

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AI HL · Probability · Poisson Distribution

The number of marking errors made by a teacher on a piece of homework follows a Poisson distribution with mean 1.6. The teacher marks one piece of homework from each of 13 pupils.

Find the probability that fewer than half of the pieces of homework contain at least one marking error.

Hint

First find the probability that a single piece of homework contains at least one marking error:

    \[ P(X\geq 1)=1-P(X=0). \]

Then consider the distribution of the number of pieces of homework, out of 13, that contain at least one error.

Solution

Let

    \[ X\sim \mathrm{Po}(1.6). \]

The probability that a single piece of homework contains at least one marking error is

    \[ P(X\geq 1)=1-P(X=0)=1-e^{-1.6}. \]

Let Y be the number of pieces of homework, out of 13, that contain at least one marking error. Then

    \[ Y\sim \mathrm{B}\left(13,1-e^{-1.6}\right). \]

Fewer than half of 13 means at most 6, so

    \[ P(Y\leq 6). \]

Using a calculator,

    \[ P(Y\leq 6)\approx 0.00739. \]

Therefore,

    \[ \boxed{0.00739}. \]

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