Introduction to Probability (Animal Science)
Where does probability come in?
Often when we are using descriptive statistics, we are calculating values from a sample which we think represents the whole population of the thing we are studying.
For example, we might take a sample of 50 mature Ayrshire cows and calculate the mean and standard deviation of their weights. We may then wish to generalise the results to the population which the sample came from, in our example this would be all the mature Ayrshire cows in the UK. When we do this, we say we are making inferences about the population using the sample data.
When we make inferences about the population, there is some doubt about how accurately our predictions match the true value(s) in the population.
For example, suppose the mean weight of our sample of 50 cows was 503kg. We might infer that the mean weight of all the mature Ayrshire cows in the UK is 503kg but we wouldn't be sure that this was the case. In fact, it probably wouldn't be exactly 503kg but we might expect that it is close to this value; the questions are 'how close?' and 'how confident are we that the true value is this close?' This is where the idea of probability can be useful. We can use probabilities to describe situations where we want to make inferences or predictions but there is some doubt about whether our inferences or predictions will match the true situation.
Definitions
- An experiment is an activity where we do not know for certain what will happen, but we will observe what happens.
- An outcome is one of the possible things that can happen.
- The sample space is the set (collection) of all possible outcomes.
- An event is a set (collection) of outcomes.
Probabilities are usually expressed in terms of fractions, decimal numbers or percentages. All probabilities are measured on a scale ranging from zero to one when using fractions or decimal numbers (zero to one hundred if we use percentages).
- An event with probability zero is an impossible event and an event with probability one is a certain event.
- Two events are said to be mutually exclusive if both cannot occur simultaneously.
- Two events are said to be independent if the occurrence of one does not affect the probability of the other occurring.
Example
Suppose we have an ordinary deck of 52 playing cards (without the jokers) and we draw one card from the deck at random and then look to see what card we have drawn. The experiment is the act of drawing a card from the deck and the outcome is the particular card we happen to draw, for example we might draw the 8 of clubs, and the sample space is the set (collection) of all the different cards in the deck.
There are lots of different possibilities for events; here are just a few:
1) We draw a red card (hearts or diamonds). There are 26 outcomes in this event because 26 of the cards are red.
2) We draw an ace. There are 4 outcomes in this event.
3) We draw the 10 of spades. There is just one outcome in this event.
4) We draw a red card which is also an ace. There are 2 outcomes in this event.
5) We draw a card that is black and a diamond. There are 0 outcomes in this event.
6) We draw a card that is red or black. There are 52 outcomes in this event.
Event 5 is an example of an impossible event: it cannot happen. Black cards are either clubs or spades and so they cannot be diamonds at the same time. Notice that there are zero outcomes in this event. Even though this cannot ever happen, we still call this an event.
Event 6 is an example of a certain event: it must happen. Whatever card we draw it must be either red or black. Notice that every outcome in the sample space (every one of the 52 cards in the deck) is in this event.
Events 2 and 3 are mutually exclusive. Either of them can occur but not both at the same time, if we draw an ace then we cannot have drawn the 10 of spades (we are only drawing one card) and vice versa.
Events 1 and 2 are independent. If we draw a red card this does not affect the probability that the card is an ace. Similarly, if we draw an ace, this does not affect the probability that the card is red.
Probability Distributions
- A variable is a characteristic which can take values which vary from individual to individual or from group to group, e.g. height, litter size, coat colour, etc.
Variables are often represented by letters of the alphabet; we often talk about the variable $X$ for example.
- A random variable is a variable which can take different values with given probabilities.
- Random variables can be discrete meaning they can only take a certain set of values e.g. litter size which is always a whole number, or continuous meaning they can take any value in a given range e.g. height.
- A probability distribution is a theoretical model which we use to calculate the probability of an event occurring.
The probability distribution tells us how likely each of the mutually exclusive events is to occur. In other words, probability distributions tell us all the values a random variable can take, along with their associated probabilities. They can be presented as an equation, a chart or a table.
For any probability distribution, the sum of the probabilities of all mutually exclusive events is always one. This is really just the same as saying that something must happen when we do our experiment: there is always an outcome.