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## Probability – the OR case

We are continuing our discussion of Probability. If you missed the introduction and the “and” case of probability, here you go. Probability – the AND case

Remember that probability is the odds or percentage of likelihood of an event happening in a given set of circumstances. The range of probability is from 0 to 1, where 0 is absolute impossibility and 1 is absolute certainty.

If you know the probability of a single event, you know the likelihood that it will occur. If there is more than one event or option involved, and you want to know the overall likelihood of a specific outcome, we need some basic math techniques to figure this out how these events work together to result in a final outcome. When looking at multiple events, we can break them into two types of combinations: “and” and “or”. If it is an “and”, both events must occur. In an “or” case, one or the other or both must occur. This post covers the “or” case.

#### “Or” Probability

If we want to consider a situation where we know the probability of two events, but we don’t care if either one occurs as long as one does, then we use the “or” case. Think of having two cars and you need to get somewhere – as long as one car or the other starts, you can get to where you need to go. There are two scenarios for an “or”. One where the events can occur (or not occur) at the same time, and the other where it is only possible for one of the events to happen in a given instance. We’ll look at that scenario first – it is where the events are mutually exclusive.

##### Mutually Exclusive Events

To be mutually exclusive, only one of the set of events can happen at any given moment. It is truly one or the other – but not both. Like a coin flip, it can be either heads or tails, but it cannot be both at the same time. The hardest part of an “or” case is determining whether or not the events are indeed mutually exclusive. If they are (we’ll get to the non-mutually exclusive events in a bit), then the math is simple addition. Note that because they are mutually exclusive, the odds always add up to 1. The coin has probability of .5 that it will land on tails and .5 that it will land on heads, added together, .5 + .5 = 1.

The equation for two mutually exclusive “or” events is:

$P(a\:or\:b) = P(a) + P(b)$

When you know the probability of the mutually exclusive events, you know they will always add up to 1.  The logic here is extremely useful and the equation helps if you only know the probability of only one of the events.  You can subtract the known probability from 1 to find the probability of the other event. If the weatherman says 80% chance of rain, you know there is 20% of not rain…

##### Non-mutually Exclusive Events

In the case of non-mutually exclusive events, the probabilities are not limited by the other events. Each event can happen regardless of the other events. This requires a little bit more math to determine the overall probability. The simplest case is with two non-mutually exclusive events. For example, I need 1 of the 2 computers at home to work to catch up on makemathagame.com or look at mathbenimble.com. If both computers work, great!, but I only need 1 to be happy. So let’s say that one computer is old and only works right about 80% of the time (a probability of .8) and the newer faster model is almost always working and has a .999 probability of getting me on the internet. What is the overall probability? We can’t just add, that would put us over 100%. The equation is:

$P(a\:or\:b) = P(a) + P(b) - P(a)P(b)$

So, Pa + Pb – PaPb is then .8 + .999 – (.8)(.999) = .9998 – the overall probability increased! Think of it this way… If the good computer works 999 time out 1000, and I happen to get the 1 time in a thousand when it doesn’t work, I still have an 80% chance of getting the other computer to work. This is why the overall probability increased – the second computer adds a backup, or redundancy. This concept is key in airplanes, cars, medical devices, cell phones… all complicated systems (and even simple systems) use this concept to make things continue to work even when some piece of the system fails.

Our computer example is a simple case with only two options to consider and the equation is a simple form of a more general equation that works for multiple events in an “or” case. All you need is the probability of each event and you can find the overall probability the system will work (called reliability). This equation is:

$P(a\:or\:b\:or\:... n) = 1 - (1 - P(a))(1 - P(b))(1 - P(n))$