. A factory employing a large number of workers finds that over a period of time , the average absentees rate is three workers per shift. Calculate the probability that in a given shift (i)exactly two workers will be absent , (ii)more than four workers will be absent .

Lacking sufficient data.

To calculate the probability in both cases, we need to use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space when these events happen with a known average rate and independently of the time since the last event.

The Poisson distribution formula is:

P(x;λ) = (e^(-λ) * λ^x) / x!

where:
- P(x;λ) is the probability of observing x events in a given interval
- λ is the average number of events that occur in that interval
- e is the base of the natural logarithm (approximately 2.71828)
- x is the number of events we are interested in

(i) To calculate the probability that exactly two workers will be absent, we need to find P(x=2;λ) where λ is the average absentee rate which is given as three workers per shift. So, λ = 3.

P(x=2;3) = (e^(-3) * 3^2) / 2!

(ii) To calculate the probability that more than four workers will be absent, we need to calculate the complementary probability of the sum of probabilities from x=0 to x=4, that is, 1 - P(x=0;λ) - P(x=1;λ) - P(x=2;λ) - P(x=3;λ) - P(x=4;λ).

P(more than 4;3) = 1 - (P(x=0;3) + P(x=1;3) + P(x=2;3) + P(x=3;3) + P(x=4;3))

Now, we can use the above formula and calculate the probabilities accordingly.