What is the probability that you will cause an explosion when filling up gas?

I like to wait inside my car when my car is filling up with gas. I do this to reset my trip odometer, stay warm, etc. One — and only one — of the local gas stations has a sign (pictured below) warning me to not get in and out of my car when fueling up. The implicit claim here is that getting in and out of my car raises my conditional probability of exploding. I as curious to see if (a) that claim was true and (b) how significant that rare risk would be if it were true, and (c) why only one gas station wants to protect me from this risk.

Snopes has some interesting data about this issue based on a PEI report [Link] The data is apparently a mess, but it looks like there were 81 fires over ~7 years, and information about 64 of these fires was available for further analysis. Most of the fires occurred in the most recent year. I assumed that maybe older fires were less likely to be recorded and therefore assumed that this represented 2 years worth of fires (~24B re-fuelings). The report implicates women in entering the car but the report’s authors don’t actually identify gender in their statistics (read the Snope article). I’ll take this into account.

It’s worth noting that men get struck by lightning way more than women. Maybe women balance this out by causing more fuel fires?

Claim (a) appears to be true. Let’s say there are 81 gas explosions in 2 years (24B refuelings). That is a base rate of 3.4×10^-9, a rare risk. This doesn’t give any insight into whether going back into one’s car is a bad idea, so let’s look at that issue further. To do so, I will assume that the 17 fires are randomly either caused by reentering or not reentering one’s car. Then, we know:

P(fire and reenter) = 1.5×10^-9

P(fire and no reenter) = 1.0×10^-9

Of course, we want to find P(fire | reenter) and P(fire | enter). To do so, let’s assume that women do 36% of all fuelings (same as the proportion of miles driven by women) and that women reenter their cars 50% of the time and men reenter 10% of the time.

P(fire | reenter) = 6.25×10^-9

P(fire | no reenter) = 1.4×10^-9

Fires in cars where someone reenters their car occur 4.5 as often as when they don’t. If we look at an extreme example where all women reenter but no men do

P(fire | reenter) = 4.22×10^-9

P(fire | no reenter) = 1.65×10^-9

Here, people that reenter (all women) cause fires 2.5 more than those that do not reenter (all men).

But those that reenter may not always cause fires at a higher rate than those that do not reenter – it all depends on how often people reenter their car. Let X = the probability that people reenter their cars while fueling:

P(fire and reenter) / X >= P(fire and no reenter) / (1-X)

This is true when X < 0.591. This appears to be true, but it's a fairly realistic number even though I bet it's a bit high. Given the rarity of fires, people that reenter probably do not cause fires at statistically significantly higher rates than those that do not reenter.

Any risk that happens about once every billion fuelings is very rare. I will not fuel my car that often over the course of my lifetime, so I can expect to escape without causing a fire.

Let’s compare the risk associated with causing an explosion due to reentering one’s car to the risk associated with dying in a car accident. To do so, let’s assume that one drives 300 miles between refuelings for an apples to apples comparison. Then, the risk of dying between refuelings is 3.3×10-6 (1.1 deaths per 100M miles driven). When compared to the risk of exploding given that you reenter, you are 528 more likely to die driving than explode when reentering your car while fueling.  Therefore, I’ll conclude that if a risk from reentering my car exists but that it’s fairly insignificant (Claim b), which explains why most gas stations are not concerned about warning me (Claim c).

Have you ever caused a fire while filling your car up with gas? Do you even know of anyone second- or third-hand who exploded while fueling?