Law of Probability

Objectives:
- Use basic terms in the language of probability.
- Work simple problems involving theoretical and empirical probability.
- Apply the law of large numbers or law of averages.
- Find probabilities related to flower colors as described by Mendel in his genetics research.
- Determine the odds in favor of an event and the odds against an event.

Deterministic vs. Random Phenomena:
- A phenomenon that can be predicted exactly with obtainable information is termed deterministic.
- Conversely, a phenomenon that cannot be predicted exactly, due to fluctuation, is termed random.
- Study of Probability: Focused on random phenomena, where the outcome cannot be known for certain but can be assessed for likelihood.

Experiment: Any observation/measurement of a random phenomenon.
Outcomes: Possible results of the experiment.
Sample Space (S): Set of all possible outcomes.
Event (E): Any subset of the sample space.
- Depicted in a Venn diagram, where the circle labeled E represents event outcomes, and the rectangle represents the sample space S.
- Favorable Outcomes: Outcomes that belong to event E. If a success is observed, the event is said to have occurred.

Probability of an Event (P): A numerical measure of the likelihood of an event occurring.
- Represented as P(E).
Calculating Probability:
- The probability of an event can be determined theoretically or empirically.

When outcomes in a sample space are equally likely, the theoretical probability of event E is calculated as:
$P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}$
Example:
- A single coin toss (sample space S = {H, T}).
- Event E (landing heads) results in:
  $P(H) = \frac{1}{2}$ with one favorable outcome, heads (H), and two total outcomes (H, T).

When an experiment is conducted to find probabilities, the value derived is termed empirical probability.
Empirical Probability Formula:
$P(E) = \frac{\text{Number of Times Event E Occurred}}{\text{Total Number of Experiments}}$

Calculating Poker Hand Probabilities:
- For a straight flush:
- Probability:
  $P = \frac{36}{2598960} = \frac{3}{216580} \approx 0.00001385$
- For four of a kind:
- Probability:
  $P = \frac{624}{2598960} = \frac{1}{4165} \approx 0.0002401$

Also known as law of averages.
As an experiment is repeated, the ratio of outcomes favors any event (E) tends to converge closer to the theoretical probability of that event.
Example of Coin Toss:
- A fair coin was tossed 35 times. Ratios of heads to total tosses were computed and plotted. Observations showed that fluctuations decreased as the number of tosses increased, aligning closer to a probability of $0.5$ (50% heads).

Repeated experiments provide an empirical probability that estimates the theoretical probability.
Increasing repetitions enhances estimate reliability. Established theoretical probabilities allow deduction of outcomes in repetitions, yielding more accurate predictions.

Mendel's Experiment: Observed characteristics in pea plants for inheriting traits, particularly flower color.
- Dominance of Traits:
- Red (R) is dominant, white (r) is recessive.
- From pure red (RR) and pure white (rr) parents' offspring (Rr), all show red, but heterozygous.
- Second generation (Rr x Rr) produces offspring with various combinations:
- Third Generation Outcomes:
  - Red: $RR$ (1 possibility)
  - Pink: $Rr$ (2 possibilities)
  - White: $rr$ (1 possibility)
- Probabilities:
  - Probability of red: $P = \frac{1}{4}$
  - Probability of pink: $P = \frac{2}{4} = \frac{1}{2}$

Definition of Odds:
- Odds compare the number of favorable outcomes to unfavorable outcomes.
For event E, if there are A favorable outcomes and B unfavorable, then:
- Odds in favor of E: $A:B$
- Odds against E: $B:A$
Conversion between Probability and Odds:
- Given $P(E) = \frac{A}{A+B}$ ,
- Odds in favor of E: $A:B$
- If odds in favor of E are $A:B$ then,
- Probability: $P(E) = \frac{A}{A+B}$
Example with Lottery:
- Odds of winning are 1:9,999.
- Probability:
  $P = \frac{1}{1+9999} = \frac{1}{10000} = 0.0001$