Probability of earning more than $500 in a week

Posted on 2012-09-12
Last Modified: 2012-09-13
Suppose over the course of a week I do various programming jobs for people.

On average each job pays $10 with a standard deviation of $1.50.

The maximum number of jobs I can do in a week is 70,

What is the probability that I will earn more than $800 in a week where I manage to do 70 jobs?

This isn't a course question, I'm just trying to learn a principle of how to do these type of questions, thanks.
Question by:purplesoup
    LVL 27

    Accepted Solution

    Assuming the required randomness (reasonable), this is a standard statistics problem. For 70 jobs average intake is $700 with standard (sd) deviation of $15. But is this really true? It is not. It overstates the sd of the 700. You should square the sd (gives the variance), add them, then take the sq root. In your case 1.5^2 = 2.25. Variance of work = 22.5, sq root of work = 4.74. You now have the usual bell curve with average 700 and sd of 4.7. You want the probability of exceeding the average by about 20 x sd. Look up the result in a table. It is VERY low, essentially zero.
    LVL 27

    Assisted Solution

    (A result further away from an average by 5 sd is usually considered to be from a different set than that used to calculate the average.
    LVL 84

    Assisted Solution

    I get a sd of $12.55, but 7.968 standard deviations from the mean is still essentially impossible.

    It would be far more likely that the tail of your distribution is actually thicker then Normal or that the pay for the various jobs are not really independent.
    LVL 27

    Assisted Solution

    oxo's sd is correct. I used a wrong multiplier. But the conclusion that the probability is essentially zero stands. A sd of 8 is VERY large
    LVL 27

    Assisted Solution

    conversely, if one takes the figure of your title ($500) the answer is that you will (almost) certainly earn more than $500.

    Author Closing Comment

    Thanks for your help - I made that example up which was why the figures were a bit rubbish - but I understand the method now - thanks again.

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