The mathematics of probability underpins the analysis of financial risk
If you toss a coin it comes down either heads or tails. Both are equally likely. We "expect" one head in every two tosses.We say the probability of a head is 1:2, or ½, or "one half", or 50%.
If you roll a dice it will come up with 1,2 3,4,5 or 6. Each result is equally likely and we say the probability of throwing a four (e.g.) is 1:6, or 1/6, or "one sixth" or 16.67%. A bookie would say that a four was "5 to 1 against" (5 chances of losing against one chance of winning).
The probability of a certainty is 1. The probability of an impossibility is zero.
Two events: chance of one or other
If two events are mutually exclusive (meaning they cannot occur together) the chance of one or other of them occurring is the sum of the probabilities. So the chance of throwing either a 2 or a 5 is 1/6 + 1/6 = 1/3.
Catch: If the events aren't mutually exclusive this doesn't work. The chance of you living another 20 years may be 80%. The chance of your wife living another 20 years may be 90%. But the chance of one or other of you living another 20 years is not 170% (which would be impossible). We have double-counted the times when both you and your wife live another 20 years - i.e. when both 'events' occur. In the case of dice, both events cannot occur. You cannot throw a 2 and a five with one throw.
Two events: chance of both
If two events are independent (so one doesn't affect the other) the chance of them both occurring is the product of the probabilities. So the chance of tossing two heads in two tosses is ½ x ½ = ¼. This fits with the intuitive answer that the outcome of two coin tosses could be head-head, head-tail, tail-head or tail-tail. So head-head is "one out of 4".
If the events aren't independent this doesn't work. The chance of your pet being a cat is 40%. The chance of your pet having a long tail is 60%. But the chance of your pet being a long-tailed cat is not 40% x 60% = 24%. It is closer to 40%, since nearly all cats have long tails.
Two events: chance of one given the other
To get the right answer to the long-tailed cat problem we need the probability that cats have long tails. If 95% of cats have long tails then the probability of your pet being a long-tailed cat is 40% x 95% = 38%. If this answer is not intuitive for you try looking at the make-up of an average population of 100 pets. We know that 60 of them will be dogs and 40 will be cats. 5% of the cats (2 of them) will have short tails. The other 38 will have long tails.
The figure of 95% is called a conditional probability. It's the probability of your pet having a long-tail conditional on your pet being a cat.
Two events: chance of the other given the one
The chance of your feline pet having a long tail is not the same as the chance of your long-tailed pet (let's limit to cats or dogs) being feline.
To answer that, we need to know something about long-tailed dogs. Suppose we know that 20% of dogs are long-tailed. Then in our average population of 100 pets our 60 dogs are made up of 12 with long tails and 48 without. The 38 long-tailed cats and 2 others we worked out before. So 38 out of 50 (38+12) long-tailed pets are cats, a probability of 76%.
Probabilities alter as we get more information
You have just had an example. The probability of your pet being a cat is 40%. But on being told that your pet is long-tailed the probability of its being a cat goes up (because most cats are long-tailed and many dogs are not). And in fact we have just worked out that the probability is 76%.
The message is......
This stuff is much harder than it looks. It is well known that people are easily confused by conditional probabilities and you can understand why. Analysts and journalists are not immune.