The Poisson distribution — common mistakes

Always be vigilant about the interval specified in the question. If the rate is given per hour and the question asks about a 5-minute interval, you MUST adjust $\lambda$ accordingly. A common mistake is to use the given rate without scaling it to the correct interval.

The Binomial distribution counts the number of 'successes' in a fixed number of trials, $n$. For example, the number of heads in 10 coin flips. The Poisson distribution counts the number of events in a fixed interval of time or space, where there is no theoretical upper limit to the number of events that can occur.

You can find an exact probability by subtracting two cumulative probabilities: $P(X=r) = P(X \le r) - P(X \le r-1)$. For example, $P(X=3) = P(X \le 3) - P(X \le 2)$.

There isn't a single definitive number, but a common rule of thumb used in A-Level Mathematics is $n > 50$ and $p < 0.1$. The quality of the approximation improves as $n$ gets larger and $p$ gets smaller. It's also generally best when the mean, $\lambda = np$, is not too large (e.g., $\lambda < 15$).

Yes, absolutely. $\lambda$ represents the average number of events in an interval, and averages can certainly be non-integers. For example, if a call centre receives 5 calls every 2 hours on average, the rate is $\lambda = 2.5$ calls per hour.