Simplified Rules to
Determine Probability
Terms:
Probability is a quantitative measure of the likelihood of a given
event. If the likelihood of an event happening is certain than it is assigned a
1. If there is no likelihood of an occurrence happening, then the probability
is assigned a 0 (zero). All other probabilities are between 0 and 1.
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.
(Image from https://en.wikipedia.org/wiki/Philippine_five_centavo_coin)
Rule # 1 Determining Probability
The probability of any outcome is
the ratio of the total number of outcomes corresponding to the event, to the
total number of outcomes.
Specific event / Total number of
possible events
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:
Getting Heads
Heads or tails
or
1 specific outcome 1
2 possible outcomes 2
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
When we want to know the
probability of the occurrence of one of two events.
The probability that at least one
of two events will occur is the sum of the probabilities of the two events,
minus the probability that both events will occur.
Sum
of the Probability of two events occurring - probability both events will occur
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
(4/52 + 13/52) - (1/52) or 17/52
- 1/52 = 16/52 or simplified, a 1/13 chance of getting an Ace or Heart
Rule # 3 Determining probability for Independent Events
When events are independent in
space and time, there is a means to determine the probability of the events
occurring.
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.
(1/2) multiplied by (1/6) = 1/12
or about .083 or 8.3%
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 is 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 = ¼
Applying techniques of probability to situations that don't obviously
call for them.
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" then
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(A) + P(Not A) = 1
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%
(Source: http://getpowers.com/probability/) Note: If you want this post remove, please email me.
