I was a bit confused about this question.
You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it to the first person offering 220 or more. Assume offers to buy the dresser are independent exponential random variables with a mean of 150. The price is firm, and you keep taking offers until you receive one that is at least 220.
What’s the expected number of rejected offers until a sale?
To solve this, I first calculated the probability that any single offer is too low. I did this by just using the CDF of an exponential with mean 1/150 and P(X < 220), equalling 0.7693. Then, I treated the expected number of rejected offers until a sale as a geometric distribution, and took the expectation of this (1 / p). However, my 1/p is really low, and it doesn't seem right.
You correctly calculated the probability of the offer being too low: if $X$ is the offered amount, then $$\Pr[X < 220] = 1 - e^{-22/15} \approx 0.769307.$$
But for a geometric distribution that counts the number $N$ of offers until the first success is observed, the parameter $p$ represents the probability of a successful offer; i.e., $$p = \Pr[X \ge 220] = e^{-22/15} \approx 0.230693.$$ Consequently, the expected number of offers will be $$\operatorname{E}[N] = 1/p = e^{22/15} \approx 4.33476.$$
And this makes intuitive sense. If the chance of a successful offer is a bit less than $1$ in $4$, then on average you'd need to observe a bit more than $4$ offers to get a successful one.
As an exercise, consider the following extension to your question. Suppose instead of accepting the first offer that is at least $220$, you instead decide that you will wait for the second offer that is at least $220$, and among these two valid offers, you will take the greater of the two. So for instance, suppose you observe the sequence of offers
$$(153, 200, 52, 100, 300, 75, 110, 250).$$
You stop accepting offers once you get $250$ since it is the second offer that is at least the minimum. But you award the dresser to the person who offered $300$.