Probability

Probability is expressed as a number between zero(impossible) and one(possible). The Value of .5 is the fifty percent chance of probability.

Probability consists of events. There are two kinds of events: Mutually Exclusive and Independent

In a Mutually Exclusive Event, there can be only one...one Event that can occur.

Should event A occur, it is impossible for event B to do so. If I flip a coin, I get heads or tails but not both. Exclusive events are calculated by adding the probabilities.

An Independent Event has no effect on other events. Where as Mutually Exclusive Events cancel the other events, Independent Events can co-exist.

Being unrelated we can say: "There was a power outage today." and "I had a hoagie for lunch." When calculating the probability of Independent Events we multiply the probabilities.

EXAMPLE
In the infamous game of backgammon - rolling a one and a six is considered one of the classical moves. How is this a mutually exclusive event? We have a pair(two) dice. We need to roll a 1 with one die and a 6 with the other. Bear in mind that it matters not if we roll a 6 or a 1 with the first die. Either will suffice. The chances of rolling either a 1 or a 6 with the first die is:

rolling a 1 = 1/6
rolling a 6 = 1/6

There are six sides to a die and we are looking to roll one number in particular. This means we have a 1 in six or 1/6 chance of rolling our number. Since we are looking for a 1 or a 6 we have:

1/6 + 1/6 = 2/6 = 1/3

There is a 1/3 chance of rolling one of two desired numbers with the first die. We are adding because if the roll is a 1 it cannot be a 6 and if it is a 6 it cannot be a 1.

Let us say we rolled a 6 with the first die.

The probability of the second die being a 1 is 1/6 as it was before, only we have already rolled one die and rolled a 6. This second die needs to be a 1.

There is a 1/6 chance of us rolling a 1 with the second die.

It is at this point that the event becomes Independent because the chances of rolling a 1 and a 6 in any order can co-exist. Here we multiply.

1/3 * 1/6 = 1/18

There is a 1/18 chance that you will roll the classic "one and a six"

Say we want to roll doubles. What are the chances of that? Because the outcome of one die is unaffected by the outcome of the other they are independent, and we multiply:

1/6 * 1/6 = 1/36 or 2.77%(1 divided by 36 times 100) chance of rolling doubles.

In examining Converse Probability we are looking at the chance something DOESN'T happen. When rolling one die what are the chances that we don't roll a 1? 1 - 1/6 = 5/6.

THE BIRTHDAY PARADOX or the BIRTHDAY PROBLEM

How many people need to be in the same room before it is more likely than not that two of the people share a birthday?

The first person enters the room and states their birthday. The second person enters. Here there is a 1/365 chance that the birthdays are the same AND a 364/365 chance that they are not. A third person enters and there being two possible matches for the third birthday, there is a 363/365 chance that there is no match. These are Independent events, so we multiply:

(364/365) * (364/365) = 0.9945297
(0.99726027 * 0.99726027)* 100 = 99.4% that two people will not share a birthday

(364/365) * (363/365) = 0.991796 or 99.2% chance that three people will not share a birthday

(363/365)*(362/365) = 0.98634636 or 98.6% chance that four people will not share a birthday

By looking at the probability of people not sharing a birthday we find that as soon as the chances are 50% that a birthday is not shared it's a also a 50% chance that a birthday is. If we continue to calculate the probability of another person entering and stating their birthday, we find that it is when the 23rd person enters we are at 50%.

Following this pattern by the time there are 100 people in the room there is a 3 million to one chance that two people share a birthday.

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