One of the most common facts said to console someone who fears flying is; “you are more likely be in a car accident on the way to the airport than be in a plane crash.” This is obviously true. The odds of being in a car accident is roughly 1/84 while the odds of being in a plane crash is in the ballpark of 1/500,000.
My mom has a relatively significant fear of flying. The times which she was forced to fly, I remember her trying to fall asleep before lift off while today she tries her hardest to avoid ever flying– I would say she hasn’t flown in the last 15 years or so. Well one day I was pondering to myself about this irrational fear (shortly before I was to be flying to Atlanta) when a thought arose. I wondered; maybe their fears aren’t completely irrational. Maybe the question that is being asked isn’t properly answering what they really fear. Clearly you are more likely to be in a car accident than a plane accident. But you are much more likely to die if you are in a plane crash rather than if you are in a car crash– By roughly 66 times. I think that what aerophobics fear most is dying by plane crash. So I decided to see if this question had any weight to it. Using my mathematical background, I knew which mathematical model could answer this question. The model is known as conditional probability.
When first taking a course on probability, conditional probability is one of the first things one begins to learn. It is actually very simple to grasp at first, though the questions can become relatively complex and most universities are aware of all of the harder questions. Lucky for us, the question we are asking today is pretty easy. First however, I need to give a quick introduction to probability.
When determining the probability of an event, questions can be quite elementary. For instance, suppose you have a bag with 4 green balls and 12 red balls. If you put your hand in the bag (and you can’t see inside) and choose a ball at random, what is the probability that you choose a green ball? The answer is simple, there are 16 balls in the bag, 4 of which are green. Therefore the probability you choose a green ball is 4/16 which reduces to 1/4. Conditional probability is just a little more complex.
The easiest way to look at conditional probability is to understand that there are just a few more variables. One of the most basic ways that conditional probability can be stated is this; given two events, A and B, with the probability of B not equal to zero. What is the conditional probability of A, given B? This may sound confusing but I believe an example will help.
Suppose you have 2 bags. Suppose bag 1 contains 4 red balls and 2 green balls, while bag 2 contains 3 red balls and 5 green balls. Suppose you are given a bag and you randomly select a ball out of the bag and it is a red ball. The question can then be asked, given that you selected a red ball, what is the probability it was from bag 2?
Now we need to know some terminology.
P(R) means: Probability of selecting a red ball. Similarly, P(G) would be “math” for “the probability of selecting a green ball”. Now the next one may be difficult if you are not mathematically savvy.
To “say” the question stated above which was; given that you selected a red ball, what is the probability you selected it from bag 2? We would write in “math” language, P(B2|R).
You can almost read that across. what is the (P)robability that you selected from B2 (bag 2) given (think of the vertical line as the word, given, here) that you selected a (R)ed ball.
Now I won’t go into why this is so since this is just a blog post and not a class, but there is a given formula that will answer this question for us which is,
Looking at this equation, I know which values that I need to determine. I need to find the probability of picking a red ball given that I have selected bag 2, I need the probability of selecting bag 2, and then I need to do the same thing, but this time, for bag 1. The solutions respectively are,
Now we Just need to plug these values into the equation above and we get,
And there we have it! We have solved a conditional probability question. Now onto the question that matters.
For more on conditional probability, Wikipedia should suffice.
The Real Question
Like I said at the beginning of this post, it’s about asking the right question. I think what those who fear flying the most are truly afraid of is death. So, to ask the question in terms of conditional probability, the question we are answering is;
Given that you die, what is the probability that you were in a plane crash versus a car crash. Written out mathematically, the question appears as follows,
Let F = fatal, A = car crash, B = plane crash. Then,
is “math” language for the question above.
We know how to answer this. We can simply follow the example above in order to solve. Now the numbers I found were from reputable sources, but if they were incorrect, that would change the results. For simplicity, we see that answering one of these questions will solve the other. So we can re-ask the question as;
Given that you die, what is the probability it was in a plane crash?
We know from our example above that,
Answering each piece of the formula in order, again by research, I found that the probability of dying given that you are in a plane crash is about 33%, the probability of being in a plane crash is 1/500,000. The probability of dying given that you are in a car accident is about 0.62%, and the probability of being in a car accident is about 1/84. Using these values, we find that…
The probability of being in a plane crash, rather than a car crash, given that you died, is…
This means that if you take 111 people from a pool of people that died in either a car accident or a plane accident, 110 of them will have died in a car crash. So yes, the answer is very underwhelming, but this should be just another fact that lets all those who have a fear of flying take a breath and relax while looking at the numbers that reassure us even more to the safeties of flying.
A quick story to share with you now that I have finished. I worked on this about a month or two ago and had found a couple different statistics, which I later discovered to be incorrect. But before I knew this, I was given chilling results that suggested the probability in dying in a plane crash versus dying in a car crash was 47.2%. Much more concerning than the unintimidating 0.9%. So lastly, breathe easy, relax, and enjoy your safe flight.