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What is an event?
any collection of results or outcomes of a procedure.
What is a simple event?
an outcome or an event that cannot be further broken down into simpler components.
What is a sample space?
a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further.
What is relative frequency approximation of probability?
- Conduct a procedure, count the number of times that event A actually occurs. Based on these results,
- P(A) = (number of times A occurred) (number of times procedure was repeated). We obtain an approximation rather than an exact value.
What is the Law of Large Numbers?
As the total number of observations increases, the corresponding approximations tend to hget closer to the actual probability.
The probability of an impossible event is ____.
The probability of an event that is certain to occur is _____.
For any event A, the probability of A is between ____ and ____ inclusive.
What is the rouding off rule of probabilities?
either give the exact fraction or decimal or round off final decimal results to three significant digits.
The ____ _____ ______ event A occurring are the ratio P(not A)
P(A), usually expressed in the form of a:b or "a to b" where a and b are integers having no common factors.
actual odds against
The ____ ___ __ _____ of event A occurring are the ratio P(A)
P(not A) usually expressed in the form of b:a or "b to a" where a and b are integers having no common factors.
actual odds in favor
How is the payoff odds calculated?
the payoff odds against even A occurring are the ratio of net profit, if you win, to the amount bet.
payoff odds against event A - (net profit):(amount bet)
If events A and B cannot occur at the same they are _______.
disjoint or mutually exclusive.
either one or the other or both is the
Either one or the other but NOT both is
Any event combining two or more simple events is calla
What are the rules of complementary events?
- P(A) + P(not A) = 1
- P(not A) = 1 - P(A)
- P(A) = 1 - P(not A)
What is the Formal Addition Rule?
- P(A or B) = P(A) + P(B) - P(A and B)
- where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure.
Two events A and B are ________ if the occurrence of one does not affect the probability of the occurrence of the other.
What is the Multiplication Rule?
- P(A and B) = P(A)P(B) if A and B are independent.
- P(A and B) = P(A)P(B|A) assuming event A has already occurred.
P(A or B), "or" suggests __________.
P(P and B), "and" suggests __________.
- Or suggests addition
- And suggests multiplication.
Probability of "at least one" is equivalent to "one or more".
How is the P( at least one) found?
- 1. use the symbol A to denote the event of getting at least one.
- 2. Let the complement of A respresent the event of getting none of the items being considered.
- 3. Calculate the probability that none of the outcomes results in the event being considered.
- 4. Sutract the result from 1. That is, evaluate
- P(at least 1) = 1 - P(none)
A conditional probabilty of an event is a probability obtained with the additional information that some other event has already occurred. P(B|A) can be found by:
dividing the probability of events A and B both occurring by the probability of event A.
For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events toge3ther can occur a total of m * n ways. This is called the _______.
Fundamental Counting Rule
A collection of n dfferent items can be arranged in order n different ways. (the first item may be selected n different ways, the second item may be selected n-1 different ways, and so on) This is the _____________ rule.
There are n different items available. We select r of the n items without replacement. We consider rearrangements of the same items to be different sequences (ABC is different from CBA). This is the _________ rule, when items are all different and is calculated by:
There are n items available, and some items are identical to others. We select all of the n items without replacement. We consider rearrangments of distinct items to be different sequences. This is the _________ rule when some items are identical to others and is calculated by:
There are n different items available. We select r of the n items (without replacement) We consider rearrangments of the same items to be the same. (ABC is the same as CBA).
This is the __________ rule and is calculated by:
When different orderings of the same items are counted separately, we have a ___________ problem, but when different orderings of the same items are not counted separately, we have a ___________ problem.
What are the five counting techniques?
- 1. fundamental counting rule
- 2. factorial rule
- 3. permutations rule(items are all different)
- 4. permutations rule (when some items are identical to others)
- 5. combinations rule
What is the formula for the Mean of a probability distribution?
What is the formula for the variance for a probability distribution? (2 versions)
The _________ of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of ___________.
The criterion for unusually high number of successes can be expressed as:___________
The criterion for unusually low number of successes can be expressed as: ___________
P(x or more )
P(x or fewer)
A binomial probability distribution results from a procedure that meets all of the following 4 requirements:
- 1. procedure has fixed number or trials.
- 2. Trials must be independent.
- 3. Each trial must have all outcomes classified into two categories (success and failure)
- 4. Probability of a success remains the same in all trials.
If a sample size is no more than ___% of the size of the population, treat the selections as being independent even if they are dependent.
What is the formula for the Binomial Probability?
- n = number of trials
- x = number of successes
- p=probability of success in anyone trial
- q=probability of failure in any one trial (q=1-p)
What is the formula for Binomial Distributions
What are the requirements for the Poisson Distribution?
- 1. random variable x is the number of occurrences of an event over some interval.
- 2. occurrences must be random
- 3. occurrences must be independent of each other.
- 4. occurrences must be uniformly distributed over the interval being used.
What is the Poisson Distribution?
a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences. The interval can be time, distance, area, volume, or some similar unit.
What is the formula for the Poisson distribution?
- where e 2.71828
How does the Poisson distribution differ from a binomial distribution? (2 ways)
- 1. Binomial distribution is affected by sample size n and a probability p, whereas Poisson distribution is affected only by the mean.
- 2. Binomial distribution, possible values of the random variable x are 0,1..n but Poisson distribution has possible x values of 0,1,2... with no upper limit.
What are the requirements for using the Poisson distribution as an approximation to the binomial?
- 1. n 100
- 2. np 10