CH 05 Probability

random experiment

A process that generates well-defined experimental outcomes. On any single repetition or trial, the outcome that occurs is determined by chance.

what is an experimental outcome of holding a stock (random experiment).

the price of the stock either increases, decreases or stays the same. For example, rolling a dice results in 1,2,3,4,5 or 6 as an experimental outcome.

What is sample space?

S, or all possible outcomes in a random experiment. As flipping a coin (random experiment) results in heads or tails as experimental outcomes, S={Heads,Tails}

what is an event

A collection of outcomes, such as from S. Example: S = {8,9} makes event S true if either condition 8 or 9 is met.

how are probabilities added

the likeley hood of each event condition occurring added, such as S = {8,9} you would find P(S)= P(8) + P(9)

What is the complement of the event

For event A, the complement of event A are all outcomes that do not fall under Evenet A’s requirements.

What is true for Events and Complements of Events?

P(A) + P(A^C) =1 (Event can be any event, such as A)

What is the union of A and B?

The event containing all outcomes being to A, to B or to both (The entire area of a Venn Diagram of two intersecting circles). A U B

What is the intersection of A and B?

The event containing outcomes that belong to both A and B only. (The Venn Diagram’s overlapping point only). A (upside down U) B

Addition Law

P(A U B) = P(A) + P(B) - P( A [upside down U] B). This is the probability that event A occurs, event B occurs, or if both occur,

Consider a study conducted by the human resources manager of a major computer software company. The study showed that 30% of the employees who left the firm within two years listed dissatisfaction with their salary as one of the reasons, 20% listed dissatisfaction with their work assignments as one of the reasons, and 12% of the former employees indicated dissatisfaction with both their salary and their work assignments. What is the probability that an employee who leaves within two years does so because of dissatisfaction with salary, dissatisfaction with the work assignment, or both?

.38, as .3 + .2 - .12 =.38. Use Addition Law

What are mutually exclsuive events?

Events that have no outcomes in common, hence P(A [Upside Down U] B) = 0

What is conditional probability?

The probability of an event given that another event has already occurred - The probability of A, given B. The conditional probability of A given B is P(A|B) = P(A (upside down U) B)/P(B)

What is a joint probabiltiy?

the probability of two events both occurring - the probability of the intersection of two events.

What are marginal probabilites?

The values in the margins of joint probability - the probabilities of each separate event.

What are independent events?

Where P(A|B) = P(A) or P(B); the two events have no influence on each other.

what is the multiplication law for dependent events?

P(A [Upside down U] B)=P(B)P(A|B). This determines the probability of the intersection of two events.

what is the multiplication law for independent events?

P (A [Upside down U] B)=P(A)P(B)

what is bayer’s thereom?

Bayes’ theorem is applicable when events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space

What is a random variable?

A numerical description of the outcome of an experiment. The value is thought to be not known with certainty,

What is a continous random variable

A random variable that may be any numerical variable in an interval or intervals, such as if my experiment is a customer visiting my website, the continuous random variable would be the time that the customer spends being greater than or equal to 0 minutes.

what is a probability distribution function?

describes the range and relative likelihood of possible values for a random variable. For a discrete random variable x, the probability distribution is defined by a probability mass function, denoted by f(x). For example, if the f(0)=.6 and f(1)=.4, we can graph these probabilities based on these values.

What is an empirical probability distribution?

A probability distribution for which the relative frequency method is used to assign probabilities.

What is a custom discrete probability distribution?

A probability distribution for a discrete random variable for which each value that the random variable assumes is associated with a defined probability. If we have limited data and good intuitions, we can use our institutions to substitute this data. If we have sufficent data, we can use relative freauencies instead.

What is the expected value?

A measure of the central location, or mean, of a random variable. E(x) = Sigma(xf(x))

What is the expected value of this data set?

13.8

How can we find E(x) using excel?

=SUMPRODUCT(x ranges,f(x) ranges)

what is a discrete uniform probability distribution?

A probability distribution in which each possible value of the discrete random variable has the same probability. F(x)=1/n, such as the probability of every side of a dice being roled is 1/6, or F(x)=1/6.

How are probabilities computed for discrete random variables?

The probability mass function f(x) provides the probability that the random variable assumes a particular value.

How are probabilities computed for continuous random varaibles?

ith continuous random variables, the counterpart of the probability mass function is the probability density function , also denoted by f(x). Probability density function does not directly provide probabilities. However, the area under the graph of f(x) corresponding to a given interval does provide the probability that the continuous random variable x assumes a value in that interval. So when we compute probabilities for continuous random variables, we are computing the probability that the random variable assumes any value in an interval. Because the area under the graph of f(x) at any particular point is zero, one of the implications of the definition of probability for continuous random variables is that the probability of any particular value of the random variable is zero.

What is a uniform probability distribution?

A continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length. The area under the curve is equal to 1, or the total probability of occuring. E(x) = (a+b)/2 while Variance = (b-a)²/12.

what is a normal probability distribution?

A continuous probability distribution in which the probability density function is bell shaped and determined by its mean and standard deviation.

What observations can be made about a normal probability distribution?

1. The entire family of normal distributions is differentiated by two parameters: the mean and the standard deviation . The mean and standard deviation are often referred to as the location and shape parameters of the normal distribution, respectively.

2. The highest point on the normal curve is at the mean, which is also the median and mode of the distribution.

3. The mean of the distribution can be any numerical value: negative, zero, or positive. Three normal distributions with the same standard deviation but three different means (−10, 0, and 20) are shown in Figure 5.22.

4. The normal distribution is symmetric, with the shape of the normal curve to the left of the mean a mirror image of the shape of the normal curve to the right of the mean.

5. The tails of the normal curve extend to infinity in both directions and theoretically never touch the horizontal axis. Because it is symmetric, the normal distribution is not skewed; its skewness measure is zero.

6. The standard deviation determines how flat and wide the normal curve is. Larger values of the standard deviation result in wider, flatter curves, showing more variability in the data. More variability corresponds to greater uncertainty. Two normal distributions with the same mean but with different standard deviations are shown in Figure 5.23.

7. Probabilities for the normal random variable are given by areas under the normal curve. The total area under the curve for the normal distribution is 1. Because the distribution is symmetric, the area under the curve to the left of the mean is 0.50 and the area under the curve to the right of the mean is 0.50.

8. The percentages of values in some commonly used intervals are as follows:

   1. 68.3% of the values of a normal random variable are within plus or minus one standard deviation of its mean.

   2. 95.4% of the values of a normal random variable are within plus or minus two standard deviations of its mean.

   3. 99.7% of the values of a normal random variable are within plus or minus three standard deviations of its mean.

What function can be used to find the area under the curve for a normal distribution function?

=NORM.DIST(Value of Interest, Mean, Standard Deviation, True). This finds the area from left to right, so for anything “above” this value, do 1-Norm.dist

What function can be used to find the area under the curve for a normal distribution function given a percentile?

=NORM.INV(Probability of Interest, Mean, Standard Deviation)

What mathematical operation is used to express when the probability of an event is equal to the probabilities of the individual outcomes in a random experiment?

Sum, as the total probability (1) equals its parts.

Define the following events for rolling a fair six-sided die. Event A is rolling an even number. Event B is rolling a number larger than 5. What is the intersection of events A and B?

The intersection of events A and B is the intersection of rolling an even number (A) that is larger than 1 (B). For a six-sided die, this means rolling a 6.

What shows the relationship between the probability of an event, A, and its complement, A^C?

P(A)=1-P(A^C)

Event A as the change in price of a share of Apple stock on a particular day. Define event B as the change in price of a share of Microsoft stock on the same day. What describes the calculation of a conditional probability related to these events?

Ex: Calculating the probability that the share price of Apple goes up given that the share price of Microsoft goes up. One needs to be contingent, or reliant on another.

Event A is rolling an even number on a six-sided die, and event B is rolling a number greater than 3. What is the P(A (Upside down U) B)?

1/3, as there are 3 outcomes for either condition Even{2,4,6} and Greater than 3{4,5,6,), which as 2/6 of them overlap, 1/3 of them apply for this INTERSECTION OF A and B.

True or False: If P(D | M) is equal to P(D), then events D and M are independent.

True, as for independent probabilities, P(A|B) = P(A) or = P(B)

The marginal probabilities in a joint probability table are found by…

Summing the joint probabilities in the row or column.

The probability of the intersection of two events is called…

A joint probability.

Why are probabilities revised in probability analysis?

To incorporate new information into the prior probability estimates for specific events

Which of the following results from revising prior probability values in probability analysis?

Posterior Probabilities, as Bayes’ Theorem helped us incorporate new information into our prior probability estimates for specific events, hence it becomes more accurate.

In probability analysis, Bayes' theorem is applicable when events to compute posterior probabilities are…

mutually exclusive and their union is the entire sample space, as a partition must form in the sample space (think of the diverging lines).

A discrete random variable is a random variable that can take on…

only specified discrete values

A continuous random variable is a random variable that…

may assume any numerical value in an interval or collection of intervals

For a continuous random variable, the probability of any specific value is…

0 - these are intervals.

A variance in probability and statistics is described as a measure of…

variability in the values of a random variable

The two parameters that define a particular normal distribution are…

the mean µ and the standard deviation σ.

what value of probability is certian/certainty?

1, or 100%

Suppose P(A)=.5, and P(B)=.6. Suppose they are independent. What is P(A|B)?

P(A|B)=P(A)=.5

What are complementary events?

An event where one outcome means the other cannot possibly happen, such as flipping a coin either being heads or tails - no both. MUTUAL EXCLUSIVITY

x bar meaning

sample mean

lowercase sigma

pop. standard deviation

s² meaning stats

sample variation

s meaning stats

sample standard deviation

lowercase mu (upsidedown h) stats meaning

population mean

N meaning stats

population size

n (lowercase) stats meaning

sample size

Is the probability for two independent events intersecting zero?

No, as they can still intersect. Dependent would mean one affects the other

What is the classical probability method?

Probability based on equally likely outcomes rather than personal judgement.