Random processes are memoryless. If you flipped a coin and turned up heads the first time. Does that increase the chances that the next coin flip will be tails to more than 50%? If the coin flip is fair, than the answer is no. The coin has an equal likelihood of getting heads or tails on the next flip. The probability remains unchanged. When a process is memoryless, like the flip of a coin, successive events are independent of each other.

Rule #1 Determining Probability

Example: What is the probability or likelihood of getting heads in a coin flip. The specific event is "Getting Heads". The Total number of events possible include getting heads or tails, two distinct possibilities. The Ratio would look like this:

Or one half or 50%. There is equal chance of a getting heads or tails on a coin flip, so the probability of getting heads is one half or 50%.

Rule #2 Law of Unions

Example: Get an Ace or a Heart from a deck of 52 cards. There are 4 Aces and 13 Hearts in a deck of cards. There is one Ace of Hearts. So the probability would be

Rule #3 Determining probability for Independent Events

For independent events, the probability of joint occurrences is equal to the product of the probabilities of the separate events. so, you can multiply their probabilities together and you will get the probability that both events will occur.

Probability of event one x Probability of event two.

Example, get a heads on the flip of a coin and roll a 6 on a 6 sided die.

Rule #4 Determining probability for Dependent Events

For dependent events, the probability of joint occurrence is equal to the product of the probability of the first event and the probability of the second event given that the first event has occurred.

For example, there are 10 marbles in a bag, 5 red and 5 blue. What is the chance I will pick two red marbles out of the bag without seeing them. I have a 5/10 chance for choosing red as the first marble and a 4/9 chance for choosing red as the second marble.

Thus my chance are 5/10 times 4/9 or a 20/90 or a 22.2% chance of choosing two red marbles.

This is called Sampling Without Replacement.

If we sample with replacement, than the probability would be 1/2 times 1/2 = 1/4th.

In situations with no clean space of equally likely possibilities
First, define a equivalence lottery (impose a kind of lottery to events that are not all equally likely. Rather than {rain, no rain}, we can use past statistics to help us create a lottery. Such as 6 days out of past three Julys it has rained. Thus there is a 6/93 chance of rain in July.

Subjective and Objective Probability

Objective Probability is a probability everyone can agree on, like fair dice. Everyone can agree that a die has a 1/6 chance of getting a 6 on a roll.

Subjective Probability - depends on the person making the assessment. A monkey, me and a stock market analyst, based on our ability to analyze or any relevant information and knowledge, may have different views on the likelihood that the market will be higher tomorrow.

The de Finetti Game by Bruno de Finetti

Say your friend says he got 100% on a test. To check his real subjective probability, you could ask him: Let's play a game. You have a choice. You can either draw a ball from a bag that has 98 red balls and two black balls. If you happen to draw a red ball, I will give you a million dollars. Or you can decide to wait to see how you did on the test. If you get a perfect score on that test, I will give you one million dollars. What's you choice: draw or wait?

If they say "draw" than you can ask how about a bag with 80 red balls and 20 black balls. So on and so on, until you get a good idea of their subjective probability.

Do not use when objective probability can be determined. In other situations, we do our best to assess our subjective probability of the outcome of an event. Once assessed it can be used with usual probability rules.

You did two interviews...
60% chance get offer from company A
20% chance get offer from company B
10% chance get offer from both

Chance you will get at least one offer is:
60% + 20% - 10% = 70%

P(Not A) = 1.00 - P(A)

The opposite of A is called the complement of A.

P(A or B) = 1 - P(Not A and Not B)

For independent events
P(A or B) = 1 - P(Not A)(Not B) meaning (Not A) and (Not B) multiplied.

Example... woman has five blind dates. She believes she has a 20% chance of having a successful relationship with each one. So her total chance of success is:

P(A or B or C or D or E) = 1.00 - P(0.8 x 0.8 x 0.8 x 0.8 x 0.8) = 0.67 or 67 percent. Pretty good odds.

So odds of getting at least one heads on two flips of a coin is:

1.00 - (1/2) x (1/2) = .75 or 75%

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