# Basics of Probability

Here are my thoughts on the basics of probability.

**Basics of Probability**

An event is any collection of results or outcomes of a procedure.

A simple event is an outcome or an event that cannot be further broken down into simpler components.

The sample space for a procedure consists of all possible events. That is, the sample space consists of all outcomes that cannot be broken down any further.

**Simple Events**

With one birth, the result of 1 girl is a simple event and the result of 1 boy is another simple event. They are individual simple events because they cannot be broken down any further.

With three births, the result of 2 girls followed by a boy is a simple event.

When rolling a single die, the outcome of 5 is a simple event, but the outcome of an even number is not a simple event.

**Simple Events and Sample Spaces**

With three births, the event of 2 girls and 1 boy is not a simple event because it can occur with different simple events.

With three births, the sample space consists of the eight different simple events.

Probability plays a central role in the important statistical method of hypothesis testing. Statisticians make decisions using data by rejecting explanations based on very low probabilities.

In probability, we deal with procedures that produce outcomes. An event is any collection of results or outcomes of a procedure. A simple event is an outcome or an event that cannot be further broken down into simpler components. The sample space for a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further.

**Notation for Probabilities**

P denotes a probability

A,B, and C denote specific events

P(A) denotes the probability of event A occurring

**Three Approaches to Finding the Probability**

Conduct a procedure and count the number of times that event A occurs. P(A) is then approximated as follows:

- Relative Frequency Approximation- \(P(A) = \frac{\text{number of time A occurred}}{\text{number of times procedure repeated}}\)
- Classical Approach to probability - If a procedure has n different sample events that are equally likely, and if event A can occur in s different ways, then: \(P(A)=\frac{\text{number of ways A occurs}}{\text{number of different simple events}}=\frac{s}{n}\)
- Subjective Probabilities-P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

**Simulations**

Sometimes none of the preceding three approaches can be used. A simulation of a procedure is a process that behaves in the same ways as the procedure itself so that similar results are produced. Probabilities can sometimes be found by using a simulation.

**Rounding Probabilities**

When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. When a probability is not a simple fraction such as \(\frac{2}{3}\), express it as a decimal so that the number can be better understood.

**Law of Large Numbers**

As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability. It tells us that relative frequency approximations tend to get better with more observations. This law reflects a simple notion supported by common sense: a probability estimate based on only a few trials can be off by a substantial amount, but with a very large number of trials, the estimate tends to be much more accurate.

Don’t make the common mistake of finding a probability value by mindlessly dividing a smaller value by a larger number. Instead, think carefully about the numbers involved and what they represent. Carefully identify the total number of items being considered.

**Complementary Events**

Sometimes we need to find the probability that an event does not occur. The complement of event A, denoted by \(\bar{A}\), consists of all outcomes in which event A does not occur.

**Identifying Significant Results**

If, under a given assumption, the probability of a particular observed event is very small and the observed event occurs significantly less than or significantly greater than what we typically expect with that assumption, we conclude that the assumption is probably not correct.

We can use probabilities to identify values that are significantly low or significantly high.

- High number of successes: x successes among n trials is a significantly high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less.
- Low number of successes: x successes among n trials is a significantly low number of successes if the probability of x or fewer successes is unlikely with a probability of 0.05 or less.

**Odds**

Expressions of likelihood are often given as odds, such as 50:1. Here are advantages of probabilities and odds:

- Odds make it easier to deal with money transfers associated with gambling.
- Probabilities make calculations easier, so they tend to be used by statisticians, mathematicians, scientists, and researchers in all fields.

In the three definitions that follow, the actual odds against and the actual odds in favor reflect the actual likelihood of an event, but the payoff odds describe the payoff amounts that are determined by gambling houses.

The actual odds against event A occurring are the ratio \(P(\bar{A}) / P(A) \), usually expressed in the form of a:b, where a and b are integers.

The actual odds in favor of event A occurring are the ratio \(P(A) / P(\bar{A}) \) which is the reciprocal of the actual odds against that event. If the odds against an event are a:b, then the odds in favor are b:a.

The payoff odds against event A occurring are the ratio of net profit(if you win) to the amount bet.

Payoff odds against event A = net profit:amount bet

If you bet $5 on the number 13 in roulette, your probability of winning is \(\frac{1}{38}\) but the payoff odds are given by the casino as 35:1

With P(13) = \({1}{38}\) and P(not 13) = \(\frac{37}{38}\), we get the actual odds against 13

= \(\frac{37/38}{1/38} or 37:1