the Poisson result and Exponential interpretation for spare part requirement analysis

I am confused with with the Poisson result and Exponential interpretation for spare part requirement analysis. I try to calculate the required number of spare parts for a disposable remove-replace item.

From a real life case; if:

1. warranty period for a part is $t=400 hr$
2. constant failure rate for the part is $\lambda=0,003534$ 1/hour so mean time to failure $MTTF=1/\lambda=283{\rm hours}$

then if I use Poisson distribution to calculate what is the required number of spares NOT to be in shortage for 400 hours for this part with at least a probability of 95%, solution is 4 pieces of spares. (the reason is:

1st x value which satisfies below inequality is 4.

$${P(x=0)\lor P(x=1)\lor P(x=2)\lor P(x=3)\lor P(x=4) }>0,95$$

)

However logicly $MTTF$ says that average time between failure events will be 283 hours so even the part fails at 1st hour then I will need 1 piece of spare.

So in this point I'm very confused since 1 and 4 are very different and has very serious consequences. (cost, customer satisfaction etc.)

I also had the graph for Reliability for this part. (${\rm Reliability}=e^{-\lambda(t)}$ )

Observing the graph also didn't help me!

Can you please explain "where and why my logic fails"? I think I've some trouble with understanding the true nature of exponential distribution and meaning of failure rate or MTTF.