Bohdan Kaflyuk / Sec 021
Probability: is the numeric value representing the chance, like hood, or possibility a particular event will occur, such as the price of a stock increasing, a rainy day, a defective product, or the outcome five in a single toss of a die.
There is 3 types of probability: A Priori, Empirical, Subjective.
A priori probability : Probability of success is based on prior kProbability of occurrence: X.
X= number of ways in which the event occurs / T= total number of possible outcomes.
Ex: A die has 6 sides. Each face has a number 1 to 6. If you roll a die what's the probability?
Solution: Each face is equally likely to occur. Since there is 6 faces, the probability of getting a face with 5 dots is 1/6.
Empirical Probability: Probability based on observed data, not on prior knowledge of a process. Surveys are often used to generate empirical probability. Examples of this type of probability are the proportion of individuals in using
Subjective Probability: Different from the other 2.ts , and 60% state that they have part time jobs , then the ratio is 6.0 for students having part time job.
Probability distribution for a discrete random variable: is a mutually exclusive listing of all the possible numerical outcomes along with the probability of occurrence of each outcome.
Random Variable: A variable whose value is determined by outcomes of a random experiment. SYMBOL used is usually X.
Discrete random Variable: A random variable whose possible values are discrete, i.e. clearly separated individual values.
Discrete Probability Distribution: A table or formula that shows the probability associated with each possible value of a discrete random variable.
Let X = # cars sold on Sunµ= MeanSunday at a particular dealership shows that the following probability distribution exists:
Probability distributionµ= E(X)=: ------->Expected Value of a Discrete Variable.
X P(x) µ= E(X) =∑ XP(X) -------------> Mean or Expected Value
1 0.1 √ σ = ¿ ----------> Variance
(¿ ¿P(X) )
2 0.1 = ∑ ¿ ------> Standard Deviation
The mean, µ, of a probability distribution is the Expected Value of its random variable.
To calculate Expected Value: Multiply each possible outcomes, X, by its corresponding probability, P(X), and then sum these products.(of a discrete random variable)
EX: (0)(0.01)+(1)(0.1)+(2)(0.2)+(3)(0.3)+(4)(0.15)........=2.8 Press: F6---> list:List1/XList: List/ Freq: List 2
Xi−E X ¿ ( )
2 2 ¿
Standard Deviation of a Discrete Random =ariable: = n List two:0.10, 0.10,0.2To get the answer press F1(Var 1)
EX. Home Mortgages mean : 2.8 / x= 1.5684
Approved per Week P(X) XP(X) [X- E(X) P(Xi)
i i i i i
0 0.10 (0)(0.10)=0(0-2.8) (0.10)= 0.784
1 0.10 (1)(0.10)=0.(1- 2.8) (0.10)= 0.32=* √ σ = √ 2.46 = 1.57
2 0.20 (2)(0.20)=0............
3 0.30 (3)(0.30)=0............. * The mean number of mortgages approved per week is 2.8, the Variance is 2.46,and the Standard Deviation is 1.57
5 0.10 (5)(0.10)=0.............
6 0.05 (6)(0.05)=0......
1.00 μ = E(X)= 2.8 σ = 2.46
Mathematical Model: is a mathematical expression that represents a variable of interest.
Binomial Distribution: is one of the most useful mathematical models. You use it when discrete random variable is the number of events of interest in a sample of N observations.
F5F5---F1 OR F2
IF you want to find a probability even an exact amount of times THEN USE Bpd (Probability)
X= # of successes / n= # of trials or sample space. / π= probability of success of each trial. / n!= n factorial(6!= 720)
EX: toss coin 6 times, find probability of getting 2nCaRl/ n choose r / order doesn't matter. n= 6 / π= 0.5 / x = 2 CALC: F5 then F5 then F1 choose variable
EX: Roll a die 10 times. Find the probability of getting EX: Survey of people in 30 and under 40 age bracket shows that 43 % of them investments in mutual funds. In a particular condominium are 18
x= # of 5's / n= 10 / π= 1/6....................CALC: x= 3 / numtriaadults in the age bracket. what is the probability that :
- at most 2 of these people will have inv. in mutual funds? : CALC: (C.D) = 4.1007E.03--->= 0.0041007
Mean of Binomial Distribution: The mean of binomial distribution is equal to simple size ,n, multiply by the probability of an event of interest , π.: (C.D)
e −A A x
Poisson Distribution: P(X)=
* P(X)= the probability of X events in an area of opportunity / A= expected number of events / e= Mathematical constant approximated by 2.71828/ X number of events( X = 0, 1 , 2 ...)
CALC: F5--->F6--->F1--->F1 for (Ppd).........." symbol instead of "A"......................Ppd calculates p(X = #)............Pcd calculates P(X ≤ #)
Normal distribution: EX: Suppose that the weights of adults are normally distributed when a mean of 170.0 lbs. and a standard deviation of 25.0 lbs.
CALC: F5-->F1--->F2 A. what is the probability that a person weights from 150 to 190 lbs.?
X= persons weight / = 170lbs / σ = 25.0 lbs.... CALC: lower =150/Upper= 1=25/ μ = 170...It
will give you the answer.
B. what is the probability that a person weights less than 140lbs?