A labor dispute has arisen concerning the distribution of 20 laborers to four di

Question

A labor dispute has arisen concerning the distribution of 20 laborers to four different construction jobs. The first job (considered to be very undesirable) required 6 laborers; the second, third, and fourth utilized 4,5, and 5 laborers, respectively. The dispute arose over an alleged random distribution of the laborers to the job that placed all 4 members of a particular ethnic group on job 1. In considering whether the assignment represented injustice, a mediation panel desired the probability of the observed event. Find the probability of the observed event if it is assumed that the laborers are randomly assigned to jobs.

Explanation / Answer

1) To assign the labourers to jobs, we must choose 6 of the 20 to do job 1, then choose 4 of the remaining 14 to do job 2, and choose 5 of the remaining 10 to do job 3. The other 5 will do job 4. You can form expressions for the number of ways each of these selections can be made. Their product is the number of ways in which the labourers can be assigned to the four jobs. It is the size of the sample space.

The number of arrangements that have the four in job 1 can calculated similarly. In this case, we first assign the four to job 1. Then, we need to choose 2 of the remaining 16 to the other spaces in job 1, choose 4 of the remaining 14 to do job 2, 5 of the remaining 10 to do job 3 and the rest do job 4.

The probability that the four are all in job 1 is the number of arrangements in which they are all in job 1 divided by the total number of arrangements.

[16!/(2!4!5!5!)] / [20!/(6!4!5!5!)]

I make this number to be 0.003 to three decimal places.

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