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Probability – the AND case

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The concepts of probabilities are so useful in the understanding of everyday life. The concepts give insight into many areas that aren’t necessarily math related. A few minutes of practice with these concepts will add to your ability to analyze many situations in your life.

Probability is the odds of a given event happening in a given set of circumstances. The odds are from 0 to 1, where 0 is absolute impossibility and 1 is absolute certainty. The odds are decimal numbers when used in the math, but can easily be converted to percentages for discussion.

If you know the probability of an event, you know the chances or likelihood that it will occur. But what 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. 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.

“And” Probability – Independent

Let’s look at the “and” first. When we have a situation where we want to find the probability of two events occurring, and we are sure that both events must occur, this is the “and” case. The math is simple for the “and” case – it is basic multiplication of the probabilities of the two individual, independent, events (the word independent is important and it means that one of the events does not have to happen before the other, they occur with or without the other event occurring – we will look at dependent events later).

The equation for two independent “and” events is:

P(a\:and\:b) = P(a) \times P(b)

For example, if you need to get cash from the bank and the probability that you will remember your wallet is about 80% (P(wallet) = .8) and the probability that you will remember to stop at the bank after work is 50% (P(stopping) = .5), and since both events must occur for you to get cash, the probability of getting cash (P(cash)) is P(wallet) x P(stopping) or .8 x .5  which is .4, or a 40% chance of you getting cash.

To account for more events, simply multiply the probability of each:

P(\text{a\:and\:b\:and\:...n}) = P(a) \times P(b) \times ... P(n)

Unless you can guarantee that events will occur 100% of the time (probability = 1), then adding more required events to the equation will lower the probability that they all work together to produce a specific result.  Each event will contribute a value less than 1, so each event will reduce the combined probability.  This is basically saying that the more complex a system is, the more likely it is to fail… We can work around this by adding redundancy, or backups, to the system with an “or” condition. (the discussion on OR is here: Probability – the OR case)

“And” Probability – Dependent, Conditional

Now let us look at conditional probability events. To be dependent, one of the events must precede the other. The second event cannot happen until after the first event – and since the first event may or may not happen (depending on its probability) – the second event may never even get a chance to happen. Even if the first event does occur, the second event may or may not happen (depending on its probability). Not too confusing, right?

Remember, we are looking for the probability that both events occur, with the additional consideration that a specified event must occur before the other is possible – like clouds before rain or like getting 2 Kings when playing cards, the first one has to come before you can get the second.

The equation, for P(b|a) (read ‘the probability of b, given a’.) is:

P(\text{b} | \text{a}) = \frac{P(a) \times P(b)}{P(a)}

So, for example, the probability of getting a ticket to a concert is only .5 (a 50% chance). If you get the ticket, you have about a 10% chance (the chance of b, given a has already occurred, or P(b|a)=.1) that you are able to get an autograph by hanging our at the stage door.

First is getting the ticket, then, and only then, will there be a chance to get the autograph.  Rearrange the equation:

P(a) \times P(b) = P(a) \times P(\text{b} | \text{a})

P(\text{ticket}) \times P(\text{autograph}) = P(\text{autograph}|\text{ticket}) \times P(\text{ticket})

Which is .1 x .5 = .05 – and this is the overall probability of you getting the ticket AND the autograph

The conditional probability is a bit more complicated and a little tricky to figure out the logic – but it can be quite useful.  Don’t miss the follow-up post on the “or” conditions of probability – what happens when you don’t care about both events, as long as one occurs…Probability – the OR case