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What is an event?
any collection of results or outcomes of a procedure.
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What is a simple event?
an outcome or an event that cannot be further broken down into simpler components.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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)
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If events A and B cannot occur at the same they are _______.
disjoint or mutually exclusive.
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either one or the other or both is the
inclusive OR
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Either one or the other but NOT both is
exclusive OR
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Any event combining two or more simple events is calla
compound event
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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)
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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.
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Two events A and B are ________ if the occurrence of one does not affect the probability of the occurrence of the other.
independent
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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.
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P(A or B), "or" suggests __________.
P(P and B), "and" suggests __________.
- Or suggests addition
- And suggests multiplication.
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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)
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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.
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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
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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.
factorial rule.
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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:
permutations rule
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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:
Permutations rule
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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:
Combinations rule
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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.
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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
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What is the formula for the Mean of a probability distribution?
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What is the formula for the variance for a probability distribution? (2 versions)
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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 ___________.
expected value
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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 ) 0.05
P(x or fewer) 0.05
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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.
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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.
5%
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What is the formula for the Binomial Probability?
- where
- 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)
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What is the formula for Binomial Distributions
1.
2.
3.
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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.
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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.
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What is the formula for the Poisson distribution?
- where e 2.71828
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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.
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What are the requirements for using the Poisson distribution as an approximation to the binomial?
- 1. n 100
- 2. np 10
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