1. “some thoughts on probability”
KIN 3030
“some thoughts on probability”

2. Introduction to Probability
Probability estimates refer to the predictions made about events based on known information.
Probabilities do not guarantee outcomes but suggest the likelihood or the chance that an event can occur.
For example, consider the toss of a “fair” coin. The probability that the toss should produce heads is “one out of two” or 1/2, however, even in one billion tosses, because each toss is independent leading to a random event, the outcome of tossing a head may never occur!

3. Probabilites are computed by determining the decimal representation of a proper fraction.
Probability values must range between 0 and 1, where:
0 represents the fraction 0 / x and
1 represents the fraction x/ x
Probabilites are computed as the proportion of the number of anticipated or favourable events which occur ... in relation to the number of events which can occur.

4. Most important, although probability estimates are exact,
and the more times a procedure occurs the more likely it is that the expected outome will be observed,
an expected outcome cannot be guaranteed.

5. A review of terminology
The Set. The set is defined as a collection of objects called elements.
for example: S = {A, B, C, D }
where: the elements of the set S are A, B, C, and D.
The subset. The subset is any collection of elements from a set.
for example: s1 = {A, D }
where: the elements of the subset s1 are A and D.
The null or empty set. The null or empty set is the set without any elements. The empty set is a subset of all sets.
for example: F = { } where: the set F has no elements.

6. number of times A can occur
The outcome or sample space. The set in which all possible outcomes or events can occur. The location in which events are observed, counted and measured.
An event. The outcomes which occur in an outcome space produce an event. An event occurs if the outcome space is not the null or empty set.
The probability that an event called “A” will occur, is written as P(A).The numerical representation of P(A) is computed from:
number of times A occurs
number of times A can occur

7. Consider a coin toss example.
To compute the probability of a head being tossed, think of the “head” as being the anticipated or favourable outcome.
Tossing a “tail” is an alternative outcome, and in the case of coin tosses, the remainder of the set of all possible outcomes.
Since there are two possible outcomes, and a head being tossed is one of the favourable or anticipated events,
we compute the probability of tossing a head as 1/2 or P(head) = 0.5.

8. Consider a dice roll example.
What is the probability (or chance) of rolling a given number?
A single die has six sides, each side with a different number from 1 to 6. Therefore, the set of all possible outcomes is: 1 die = {1, 2, 3, 4, 5, 6} and the probability of rolling any “given number” is 1/6 or P(roll) =
Therefore, for a single die, estimate the probability of rolling a number less than “5”. ....

9. 1 roll of a single die = {1, 2, 3, 4, 5, 6} = 6
For a single die, estimate the probability of rolling a number less than “5”.
Step 1: determine the set of all possible outcomes.
1 roll of a single die = {1, 2, 3, 4, 5, 6} = 6
Step 2: determine the set of favourable outcomes.
numbers less than 5 = {1, 2, 3, 4} = 4
Step 3: divide the number of favourable or anticipated outcomes by the number of possible outcomes to estimate the probability.
probability = 4/6 = 67%

10. Mutually exclusive events.
Sets which are mutually exclusive refer to sets in which no common elements exist.
for example,
consider two sets: SA and SB,
given that
SA contains specific elements that
DO NOT EXIST in SB
we then say that
SA and SB are mutually exclusive SETS.

11. consider two sets: SA and SB,
An intersection.
The intersection also refers to the membership of elements in sets.
An intersection exists between sets when an “outcome” or an event must be a member of the subsets which are interacting within a sample space.
for example,
consider two sets: SA and SB,
given that SA contains specific elements that also have membership in SB we then say that an intersection exists between SA and SB.
The intersection between two sets is written SA  SB.

12. The union refers to the membership of elements in sets.
A union exists between sets when an “outcome” or an event may be a member of any subset or all the subsets within a sample space.
For example, consider two sets: SA and SB, given that
SA may contain elements that have membership in SB,
we can say that a union exists between SA and SB.
The union between two sets is written SA  SB

13. The complement.
Given 2 subsets, the complement refers to the set of elements that exist in one subset but not in the remaining subset (i.e. the complementary subset).
For example,
S = {A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, D3}
where: the elements of the subset S1 are
{ A1, A2, A3, B1, B2, B3, C1, C2, C3, },
and the elements of the subset S2 are
{ D1, D2, D3},
we say that subset S2 is the complement to S1.

14. Conducting an experiment is a process involving uncertain events.
An uncertain event.
Uncertain events refer to processes, elements, or events which may lead to an outcome that is measureable but cannot be expected.
Some examples of uncertain events include:
i) selecting a head to appear from the toss of a coin,
ii) choosing six numbers on the “lotto 649”,
iii) selecting the horse “GumShoes” to place second in a horse race.
Conducting an experiment is a process involving uncertain events.

15. “if a relative age effect exists in ice hockey”
The Ice Hockey Example
While dice are great for demonstrating probability theories, few research studies are concerned with outcomes from rolling dice.
In the following example, the researchers were asked to determine:
“if a relative age effect exists in ice hockey”

16. The relative age effect refers to the selection of individuals to sports teams based on physical and psychological maturity which can be biased by month of birth.

17. For example, the current age determination date of December 31, appears to favor players who are born earlier in the calendar year (January to April), and discriminate against players born in the latter part of the calendar year (September to December),
Therefore an ice hockey player born in the first four months of any given year is likely to have an advantage over a player born in the latter part of the year.
This age effect is most obvious during the critical years of growth and development (ages 9 to 15).

18. To investigate whether a relative age effect exists in ice hockey, the researchers reviewed month of birth data from several Canadian ice hockey leagues with the expectation that:
if no relative age effect exists then the proportion of births in any month would be computed by:
The total number of births divided by the number of months in a year (12).
so that under an unbiased assumption we expect that 8.3% of yearly births should occur in each month.

19. Under an unbiased assumption we expect that 8
Under an unbiased assumption we expect that 8.3% of yearly births should occur in each month.
In the remaining slides we will use the following data set to illustrate the concepts of probability estimates as they are applied to FREQUENCY data.
The data represent the month of birth for 2047 participants in the C.I.A.U. roster. The data were used in the research study of “age determination” (Montelpare, Scott, and Pelino, 1997).

20. Month of Expected Observed Cumulative
Birth Proportion Proportion* Proportion *
January = 8.33% = 11.0%
February = 8.33% = 10.6%
March = 8.33% = 9.9%
April = 8.33% = 10.8% 42.3
May = 8.33% = 9.8%
June = 8.33% = 7.9% 60.0**
July = 8.33% = 7.6%
August = 8.33% = 7.0% 32.2
September = 8.33% = 8.0%
October = 8.33% = 6.0%
November = 8.33% = 6.6%
December = 8.33% = 4.5% 25.1
* excludes rounding error sample size of 2047
** Cumulative Proportion for 1st half of year.

21. These data are useful in demonstrating some of the characteristics of probability and set theory.
That is, we can use these data to estimate sets of probabilities or relative chance associated with specific outcomes.
A set of probabilities refers to the probabilities associated with an event or set of events, and the complement(s) of the associated probability(ies).

22. born between the months of January and April (inclusive);
For example, what is the probability that an individual within the data set for ice hockey players will be:
born between the months of January and April (inclusive);
or in the first six months of any year;
or in the last quarter of any year.

23. Probabilities associated with specific ranges of the month of birth.
1. Set of Probability Estimates
P(born 1st half of year) = 1229/2047 = 0.6 = 60.0%
P(born 2nd half of year) = 818/2047 = 0.4 = 40.0%
2. Set of Probability Estimates
P(born 1st quarter of year) = 645/2047 = = 31.5%
P(born 2nd quarter of year) = 584/2047 = = 28.5%
P(born 3rd quarter of year) = 463/2047 = = 22.6%
P(born 4th quarter of year) = 350/2047 = = 17.1%
3. Set of Probability Estimates
P(born 1st third of year) = 866/2047 = = 42.3%
P(born 2nd third of year) = 662/2047 = = 32.3%
P(born 3rd third of year) = 514/2047 = = 25.1%

24. These data describe mutually exclusive outcomes.
For example, an individual born between January and April cannot belong to the set of individuals born in the fourth quarter (October-December) of any given year.

25. Further, for any set of probability estimates (where the sum of the probabilities in the set = 1.0)
the probabilities associated with the events not selected refer to the complement probabilities.
So that the complement to the probability of being born in the first half of the year is the probability of being born in the second half of the year.

26. We can also use these data to arrange comparisons of sets of probabilities, that is to select elements which are not mutually exclusive events.
Consider the question, “what is the probability of being born in the 1st third of a year, and the 1st quarter of a year, and the 1st half of a year?”.
The solution to this question is described as follows:

27. elements in the set to compute the P(born 1st quarter of year) are number of births in {January, February, March}
elements in the set to compute the P(born 1st third of year) are number of births in {January, February, March, April}
elements in the set to compute the P(born 1st half of year). are number of births in {January, February, March, April, May, June}

28. {births in January, births in February, births in March}.
We describe the association between these probabilities in terms of an intersection as follows:
P(born 1st third of year)  P(born 1st quarter of year)  P(born 1st half of year) or
P(1) P(2)  P(3).
and the elements that belong to the outcome space of the intersection are described as:
P(1)  P(2)P(3) =
{births in January, births in February, births in March}.

29. Therefore, we could compute the probability of this intersection as:
P((1)(2)  (3)) =number of elements in ((1)  (2)  (3))
Number of elements in the possible sample space
= { } = {645} = 0.315
2047 births per year

30. Similarly, we could use these data to describe a union of probabilities. Consider the question: “what is the probability of being born in the 1st third of a year, or the 1st quarter of a year, or the 1st half of a year?”.
We describe the association between the probabilities in terms of a union as follows:
P(born 1st third of year)  P(born 1st quarter of year)  P(born 1st half of year) or P(1)  P(2)  P(3).
and the elements that belong to the outcome space of the union are described as:

31. P(1)  P(2)  P(3) = {births in January, births in February, births in March} or {births in April} or {births in May, births in June}.
Therefore, we could compute the probability of this union as:
P((1)  (2)  (3)) = number of elements in ((1)(2)  (3))
number of elements in the possible sample space
= { } ={1229}= 0.60
2047 births per year

32. Finally, we could use the month of birth data to describe two events which are not mutually exclusive but may demonstrate properties of union and intersection.
Consider the question: “What is the probability of belonging to the set “being born in the 1st half of a year” or being belonging to the set “born in the month of June”?”.
The elements of the response set are not mutually exclusive, in that the element of the set “born in the month of June” is included in the set “born in the 1st half of a year”.

33. We describe the association between these probabilities in terms of a union as follows:
P(born 1st half of year) P(born in the month of June)
or P(1) P(2)
and the elements that belong to the outcome space of the union are described as:
P(1)  P(2) = {births in January, births in February, births in March, births in April, births in May, births in June } or {births in June}.

34. P((1)  (2)) = P(1) + P(2) - P((1) (2))
In this example, we compute the probability of the union by the addition of the probabilities
P((1)  (2)) = P(1) + P(2)
but we subtract the intersection component P((1)  (2)) so that the complete formula is:
P((1)  (2)) = P(1) + P(2) - P((1) (2))

35. The probability of the union of the sets of “being born in the 1st half of the year” OR “being born in the month of June”, is equal to the probability of “being born in the 1st half of the year” plus the probability of “being born in the month of June” minus the probability of “being born in the 1st half of the year” AND “being born in the month of June”.
P((1) (2)) =
{ } - {162}
2047 births per year
={0.6} - {0.08} = 0.52

36. FINALLY, when we are considering the probability or chance of any element in a set of mutually exclusive events to occur, we ADD the probabilities associated with the independent elements in the set of mutually exclusive events.
However, if we were considering the probability or chance for all the elements in a set of mutually exclusive events to occur then we would MULTIPLY the probabilities associated with the independent elements in the set of mutually exclusive events.