How do I solve this problem on Poisson approximation to Binomial distribution?

Question

The lifetime (in years) of a bulb is distributed as an $Exp(1)$ random variable. Using Poisson approximation to Binomial distribution, the probability that out of 50 randomly chosen bulbs, at most 1 fails within one month is ____?

Answer 1

• The probability that a specific bulb fails within a month is $\int_0^{1/12} e^{-t} \, dt = 1 - e^{-1/12}$; call this quantity $p$.
• Out of 50 bulbs, the number of bulbs that fails within a month is a $\text{Binomial}(50, p)$ random variable. You want the probability that this random variable is at most $1$. Instead of using this binomial distribution directly, you are asked to use the Poisson approximation. Which Poisson distribution approximates the $\text{Binomial}(50, p)$ distribution?