The lifetime of a battery has the exponential distribution with mean 6.9 months. If it fails, I immediately replace it with another - if I have one. I need a year's supply of the battery. In addition to the one in the flashlight, how many extra should I bring with me? 1, 2, or 3?

You should carry 2 other batteries.

Step-by-step explanation:

If the average life of each battery tends to last 6.9 months, and what you need is a year of battery supply, you probably need to carry two batteries.

The battery inside the flashlight has probably already spent much of its half-life.

We are told that the flashlight already has a battery inside, the batteries are discharged inside the flashlight even if the flashlight is not being used.

Which means that the average life of that battery is wearing out.

It would take 2 more batteries in addition to the one already inside the flashlight to supply the need for these for a year.