Understanding the problem:
The problem states that the average number of murders in Delhi per week is three, and the occurrences of murders follow a Poisson distribution. We need to find the probability of having five or more murders in a given week.

Poisson distribution:
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. It is often used to model the number of events happening within a specific time period.

In this case, the average number of murders per week is three, which means λ (lambda) = 3.

Calculating the probability:
To find the probability of having five or more murders in a given week, we need to calculate the cumulative probability of having less than five murders and subtract it from 1.

The formula for the Poisson distribution is:

P(X=k) = (e^(-λ) * λ^k) / k!

Where:
- P(X=k) is the probability of having k events occur
- e is the base of the natural logarithm
- λ is the average number of events

Calculating the cumulative probability:
To calculate the cumulative probability of having less than five murders, we sum up the individual probabilities of having 0, 1, 2, 3, and 4 murders.

P(X<5) =="" p(x="0)" +="" p(x="1)" +="" p(x="2)" +="" p(x="3)" +="" p(x="">

Using the Poisson distribution formula, we can calculate these probabilities:

P(X=0) = (e^(-3) * 3^0) / 0! = (1 * 1) / 1 = 1
P(X=1) = (e^(-3) * 3^1) / 1! = (0.04978706837 * 3) / 1 = 0.1493612051
P(X=2) = (e^(-3) * 3^2) / 2! = (0.04978706837 * 9) / 2 = 0.2240418076
P(X=3) = (e^(-3) * 3^3) / 3! = (0.04978706837 * 27) / 6 = 0.2240418076
P(X=4) = (e^(-3) * 3^4) / 4! = (0.04978706837 * 81) / 24 = 0.1680313557

Now, we can calculate the cumulative probability:

P(X<5) =="" 1="" +="" 0.1493612051="" +="" 0.2240418076="" +="" 0.2240418076="" +="" 0.1680313557="">

Calculating the probability of five or more murders:
To find the probability of having five or more murders, we subtract the cumulative probability of having less than five murders from 1:

P(X>=5) = 1 - P(X<5) =="" 1="" -="" 0.765476175="">

Therefore, the probability of having five or more murders in a given week is 0.