Stats text 05

5.1 Ways to determine a probability

A probability is the likelihood of an event or outcome. Probabilities are specified mathematically by a number between 0 and 1 including 0 or 1.

• 0 is no likelihood an event will occur.

• 1 is absolute certainty an event will occur.

• 0.5 is an equal likelihood of occurrence or non-occurrence.

• Any value between 0 and 1 can occur.

We use the notation P(eventLabel) = probability to report a probability.

There are three ways to assign probabilities.

• Intuition or subjective estimate

• Equally likely outcomes

• Relative Frequencies

Intuition

Intuition/subjective measure. An educated best guess. Using available information to make a best estimate of a probability. Could be anything from a wild guess to an educated and informed estimate by experts in the field.

Equally Likely Events or Outcomes

Equally Likely Events: Probabilities from mathematical formulas

In the following the word "event" and the word "outcome" are taken to have the same meaning.

Probabilities versus Statistics

The study of problems with equally likely outcomes is termed the study of probabilities. This is the realm of the mathematics of probability. Using the mathematics of probability, the outcomes can be determined ahead of time. Mathematical formulas determine the probability of a particular outcome. All measures are population parameters. The mathematics of probability determines the probabilities for coin tosses, dice, cards, lotteries, bingo, and other games of chance.

This course focuses not on probability but rather on statistics. In statistics, measurement are made on a sample taken from the population and used to estimate the population's parameters. All possible outcomes are not usually known. is usually not known and might not be knowable. Relative frequencies will be used to estimate population parameters.

Calculating Probabilities

Where each and every event is equally likely, the probability of an event occurring can be determined from

probability = ways to get the desired event ÷ total possible events

or

probability = ways to get the particular outcome ÷ total possible outcomes

Dice and Coins

Binary probabilities: yes or no, up or down, heads or tails

P(head on a penny) = one way to get a head ÷ two sides = 1/2 = 0.5 or 50%

That probability, 0.5, is the probability of getting a heads or tails prior to the toss. Once the toss is done, the coin is either a head or a tail, 1 or 0, all or nothing. There is no 0.5 probability anymore.

Over any ten tosses there is no guarantee of five heads and five tails: probability does not work like that. Over any small sample the ratios of expected outcomes can differ from the mathematically calculated ratios.

Over thousands of tosses, however, the ratio of outcomes such as the number of heads to the number of tails, will approach the mathematically predicted amount. We refer to this as the law of large numbers.

In effect, a few tosses is a sample from a population that consists, theoretically, of an infinite number of tosses. Thus we can speak about a population mean μ for an infinite number of tosses. That population mean μ is the mathematically predicted probability.

Population mean μ = (number of ways to get a desired outcome) ÷ (total possible outcomes)

Dice: Six-sided

A six-sided die. Six sides. Each side equally likely to appear. Six total possible outcomes. Only one way to roll a one: the side with a single pip must face up. 1 way to get a one/6 possible outcomes = 0.1667 or 17%

P(1) = 0.17

Dice: Four, eight, twelve, and twenty sided

The formula remains the same: the number of possible ways to get a particular roll divided by the number of possible outcomes (that is, the number of sides!).

Think about this: what would a three sided die look like? How about a two-sided die? What about a one sided die? What shape would that be? Is there such a thing?

Two dice

Ways to get a five on two dice: 1 + 4 = 5, 2 + 3 = 5, 3 + 2 = 5, 4 + 1 = 5 (each die is unique). Four ways to get/36 total possibilities = 4/36 = 0.11 or 11% Note that there are 36 total possibilities, which is six squared. There is a formula for coins and dice to obtain the total number of possibilities: (number of sides)^(number of coins or dice). The ^ symbol is exponentiation. 6^2 is 36, so 36 unique combinations. I know this is confusing: but try using a white die and a black die and you can see that ⚀⚁ is different from ⚁⚀.

5.2 Sample space

The sample space set of all possible outcomes in an experiment or system.

Bear in mind that the following is an oversimplification of the complex biogenetics of achromatopsia for the sake of a statistics example. Achromatopsia is controlled by a pair of genes, one from the mother and one from the father. A child is born an achromat when the child inherits a recessive gene from both the mother and father.

A is the dominant gene
a is the recessive gene

A person with the combination AA is "double dominant" and has "normal" vision.
A person with the combination Aa is termed a carrier and has "normal" vision.
A person with the combination aa has achromatopsia.

Suppose two carriers, Aa, marry and have children. The sample space for this situation is as follows:

The above diagram of all four possible outcomes represents the sample space for this exercise. Note that for each and every child there is only one possible outcome. The outcomes are said to be mutually exclusive and independent. Each outcome is as likely as any other individual outcome. All possible outcomes can be calculated. the sample space is completely known. Therefore the above involves probability and not statistics.

The probability of these two parents bearing a child with achromatopsia is:

P(achromat) = one way for the child to inherit aa/four possible combinations = 1/4 = 0.25 or 25%

This does NOT mean one in every four children will necessarily be an achromat. Suppose they have eight children. While it could turn out that exactly two children (25%) would have achromatopsia, other likely results are a single child with achromatopsia or three children with achromatopsia. Less likely, but possible, would be results of no achromat children or four achromat children. If we decide to work from actual results and build a frequency table, then we would be dealing with statistics.

The probability of bearing a carrier is:

P(carrier) = two ways for the child to inherit Aa/four possible combinations = 2/4 = 0.50

Note that while each outcome is equally likely,there are TWO ways to get a carrier, which results in a 50% probability of a child being a carrier.

5.3 Relative Frequency

Note to instructors: This book is unapologetically frequentist in its approach. Hard core long term frequency is the probability of an event. The author is keenly aware of the Bayesian objection to this definition and approach to statistics. This is an introductory first statistics course for students who have not studied any statistics prior to this class. The intent is to bring them up to speed in frequentist statistics leading in confidence intervals and two-tailed hypothesis testing and then have them use these tools in data exploration exercises. This text deliberately synonymizes relative frequency with probability.

The third way to assign probabilities is from relative frequencies. Each relative frequency represents a probability of that event occurring for that sample space. Body fat percentage data was gathered from 58 females. The data had the following characteristics:

sample size n (count): 59
mean: 28.7
Standard deviation sx 7.1
min 15.6
max 50.1

Note that the classes are not equal width in this example.

This means there is a...

• 0.05 (five percent) probability of a female student in the sample having a body fat percentage between 12 and 20 (athletically fit)

• 0.25 (25%) probability of a female student in the sample has body fat percentage between 20.1 (the Tanita unit only measured to the nearest tenth) and 24 (physically fit)

• 0.41 (41%) probability of a female student in the sample has body fat percentage between 24.1 and 31 (acceptable but not fit level of fat)

• 0.20 (20%) probability of a female student in the sample has body fat percentage between 31.1 and 39 (on the borderline between acceptable and obese)

• 0.08 (8%) probability of a female student in the sample has body fat percentage between 39.1 and 51 (medically obese)

The most probable result (most likely) is a body fat measurement between 24.1 and 31 with a 41% probability of a student being in each of either of these intervals.

Remember that...

• The sum of the frequencies is the sample size n.

• The sum of the relative frequencies is always one: probabilities add to one, which is also 100%.

• The sum of the frequencies being the sample size and the sum of the relative frequencies were ways to check the accuracy of the frequency table.

Law of Large Numbers

For relative frequency probability calculations, as the sample size increases the probabilities get closer and closer to the true population parameter (the actual probability for the population). Bigger samples are more accurate.