Probability of 4 or fewer errors in 100,000 messages The probability of an error occurring in a message is 10^-5. The probability is independent for different messages. There are 100,000 messages sent. What is the probability that 4 or fewer errors occur?

In principle, the number $X$ of errors in $100000$ messages has binomial distribution.  But in this kind of situation (probability $p$ of an "error" small, number $n$ of trials large, $np$ of moderate size) it is standard to approximate the distribution of $X$ by using the Poisson distribution with parameter $\lambda=np$.
In our case we have $\lambda=np=(10^{-5})(100000)=1$. The probability of $4$ or fewer errors is approximately 
$$\sum_{k=0}^4 e^{-1} \frac{1^k}{k!}.$$ 
