A listing of all outcomes of an experiment and the probability associated with each outcome. A probability distribution gives the entire range of values that can occur based on an experiment. A probability distribution is similar to a relative frequency distribution. However, instead of describing the past, it describes a likely future event. For example, a drug manufacturer may claim a treatment will cause weight loss for 80 percent of the population. A consumer protection agency may test the treatment on a sample of six people. If the manufacturer's claim is true, it is almost impossible to have an outcome where no one in the sample loses weight and it is most likely that five out of the six do lose weight.
CHARACTERISTICS OF A PROBABILITY DISTRIBUTION
- 1. The probability of a particular outcome is between 0 and 1 inclusive.
- 2. The outcomes are mutually exclusive events.
- 3. The list is exhaustive. So the sum of the probabilities of the various events is equal to 1.
How can we generate a probability distribution?
Suppose we are interested in the number of heads showing face up on three tosses of a coin. This is the experiment. The possible results are: zero heads, one head, two heads, and three heads. What is the probability distribution for the number of heads?
There are eight possible outcomes. A tail might appear face up on the first toss, another tail on the second toss, and another tail on the third toss of the coin. Or we might get a tail, tail, and head, in that order. We use the multiplication formula for counting outcomes (5–8). There are (2)(2)(2) or 8 possible results.
- Note that the outcome “zero heads” occurred only once, “one head” occurred three times, “two heads” occurred three times, and the outcome “three heads” occurred only once. That is, “zero heads” happened one out of eight times. Thus, the probability of zero heads is one-eighth, the probability of one head is three-eighths, and so on. The probability distribution is shown in Table 6-1. Because one of these outcomes must happen, the total of the probabilities of all possible events is 1.000. This is always true. The same information is shown in Chart 6-1.
- Refer to the coin-tossing example in Table 6-1. We write the probability of x as P(x). So the probability of zero heads is P(0 heads) = .125, and the probability of one head is P(1 head) = .375, and so forth. The sum of these mutually exclusive probabilities is 1; that is, from Table 6-1, 0.125 + 0.375 + 0.375 + 0.125 = 1.00.
- In any experiment of chance, the outcomes occur randomly.
- For example, rolling a single die is an experiment: any one of six possible outcomes can occur. Some experiments result in outcomes that are quantitative (such as dollars, weight, or number of children), and others result in qualitative outcomes (such as color or religious preference). Each value of the random variable is associated with a probability to indicate the chance of a particular outcome.
- If we count the number of employees absent from the day shift on Monday, the number might be 0, 1, 2, 3,… The number absent is the random variable.If we weigh four steel ingots, the weights might be 2,492 pounds, 2,497 pounds, 2,506 pounds, and so on. The weight is the random variable.If we toss two coins and count the number of heads, there could be zero, one, or two heads. Because the number of heads resulting from this experiment is due to chance, the number of heads appearing is the random variable.
- Other random variables might be the number of defective lightbulbs produced in an hour at the Cleveland Company Inc., the grade level (9, 10, 11, or 12) of the members of the St. James girls' basketball team, the number of runners in the Boston Marathon for the 2010 race, and the daily number of drivers charged with driving under the influence of alcohol in Texas.
The following diagram illustrates the terms experiment, outcome, event, and random variable. First, for the experiment where a coin is tossed three times, there are eight possible outcomes. In this experiment, we are interested in the event that one head occurs in the three tosses. The random variable is the number of heads. In terms of probability, we want to know the probability of the event that the random variable equals 1. The result is P(1 head in 3 tosses) = 0.375. (K)A random variable may be either discrete or continuous.
Discrete Random Variable
A discrete random variable can assume only a certain number of separated values. If there are 100 employees, then the count of the number absent on Monday can only be 0, 1, 2, 3,…, 100. A discrete random variable is usually the result of counting something.
A discrete random variable can, in some cases, assume fractional or decimal values. These values must be separated—that is, have distance between them. As an example, the scores awarded by judges for technical competence and artistic form in figure skating are decimal values, such as 7.2, 8.9, and 9.7. Such values are discrete because there is distance between scores of, say, 8.3 and 8.4. A score cannot be 8.34 or 8.347, for example
Continuous Random Variable
On the other hand, if the random variable is continuous, then the distribution is a continuous probability distribution. If we measure
something such as the width of a room, the height of a person, or the pressure in an automobile tire, the variable is a continuous random variable
. It can assume one of an infinitely large number of values, within certain limitations. As examples:
- The times of commercial flights between Atlanta and Los Angeles are 4.67 hours, 5.13 hours, and so on. The random variable is the time in hours.
- Tire pressure, measured in pounds per square inch (psi), for a new Chevy Trailblazer might be 32.78 psi, 31.62 psi, 33.07 psi, and so on. In other words, any values between 28 and 35 could reasonably occur. The random variable is the tire pressure.
Difference between a random variable and a probability distribution.
- Logically, if we organize a set of possible values from a random variable into a probability distribution, the result is a probability distribution.
- So what is the difference between a probability distribution and a random variable?
- A random variable reports the particular outcome of an experiment.
- A probability distribution reports all the possible outcomes as well as the corresponding probability.
How to tell the difference between a discrete or continuous distribution.
- Usually a discrete distribution is the result of counting something, such as:
- The number of heads appearing when a coin is tossed 3 times.
- The number of students earning an A in this class.
- The number of production employees absent from the second shift today.
- The number of 30-second commercials on NBC from 8 to 11 P.M. tonight.
- Continuous distributions are usually the result of some type of measurement, such as:
- The length of each song on the latest Linkin Park CD.
- The weight of each student in this class.
- The temperature outside as you are reading this book.
- The amount of money earned by each of the more than 750 players currently on Major League Baseball team rosters.
Binomial Probability Distribution
Mutually exclusive, discrete, probability remains the same from one trial to the next, each trial is independent of other trials (no pattern)
The binomial probability distribution is a widely occurring discrete probability distribution. One characteristic of a binomial distribution is that there are only two possible outcomes on a particular trial of an experiment. For example, the statement in a true/false question is either true or false. The outcomes are mutually exclusive, meaning that the answer to a true/false question cannot be both true and false at the same time. As other examples, a product is classified as either acceptable or not acceptable by the quality control department, a worker is classified as employed or unemployed, and a sales call results in the customer either purchasing the product or not purchasing the product. Frequently, we classify the two possible outcomes as “success” and “failure.” However, this classification does not imply that one outcome is good and the other is bad.
Another characteristic of the binomial distribution is that the random variable is the result of counts. That is, we count the number of successes in the total number of trials. We flip a fair coin five times and count the number of times a head appears, we select 10 workers and count the number who are over 50 years of age, or we select 20 boxes of Kellogg's Raisin Bran and count the number that weigh more than the amount indicated on the package.A third characteristic of a binomial distribution is that the probability of a success remains the same from one trial to another. Two examples are:The probability you will guess the first question of a true/false test correctly (a success) is one-half. This is the first “trial.” The probability that you will guess correctly on the second question (the second trial) is also one-half, the probability of success on the third trial is one-half, and so on.If past experience revealed the swing bridge over the Intracoastal Waterway in Socastee was raised one out of every 20 times you approach it, then the probability is one-twentieth that it will be raised (a “success”) the next time you approach it, one-twentieth the following time, and so on. p. 196The final characteristic of a binomial probability distribution is that each trial is independent of any other trial. Independent means that there is no pattern to the trials. The outcome of a particular trial does not affect the outcome of any other trial. Two examples are:A young family has two children, both boys. The probability of a third birth being a boy is still .50. That is, the gender of the third child is independent of the other two.Suppose 20 percent of the patients served in the emergency room at Waccamaw Hospital do not have insurance. If the second patient served on the afternoon shift today did not have insurance, that does not affect the probability the third, the tenth, or any of the other patients will or will not have insurance.
How Is a Binomial Probability Computed?
- To construct a particular binomial probability, we use
- (1) the number of trials and
- (2) the probability of success on each trial. For example, if an examination at the conclusion of a management seminar consists of 20 multiple-choice questions, the number of trials is 20. If each question has five choices and only one choice is correct, the probability of success on each trial is .20. Thus, the probability is .20 that a person with no knowledge of the subject matter will guess the answer to a question correctly. So the conditions of the binomial distribution just noted are met
Hypergeometric Probability Distribution
Recall that one of the criteria for the binomial distribution is that the probability of success remains the same from trial to trial. Because the probability of success does not remain the same from trial to trial when sampling is from a relatively small population without replacement, the binomial distribution should not be used. Instead, the hypergeometric distribution is applied. Therefore,
(1) if a sample is selected from a finite population without replacement and
(2) if the size of the sample n is more than 5 percent of the size of the population N, then the hypergeometric distribution
is used to determine the probability of a specified number of successes or failures. It is especially appropriate when the size of the population is small.
- For the binomial distribution to be applied, the probability of a success must stay the same for each trial. For example, the probability of guessing the correct answer to a true/false question is .50. This probability remains the same for each question on an examination. Likewise, suppose that 40 percent of the registered voters in a precinct are Republicans. If 27 registered voters are selected at random, the probability of choosing a Republican on the first selection is .40. The chance of choosing a Republican on the next selection is also .40, assuming that the sampling is done with replacement, meaning that the person selected is put back in the population before the next person is selected.
- Most sampling, however, is done without replacement. Thus, if the population is small, the probability for each observation will change. For example, if the population consists of 20 items, the probability of selecting a particular item from that population is 1/20. If the sampling is done without replacement, after the first selection there are only 19 items remaining; the probability of selecting a particular item on the second selection is only 1/19. For the third selection, the probability is 1/18, and so on. This assumes that the population is finite—that is, the number in the population is known and relatively small in number. Examples of a finite population are 2,842 Republicans in the precinct, 9,241 applications for medical school, and the 18 2010 Dakota 4x4 Crew Cabs at Helfman Dodge Chrysler Jeep in Houston, TX.
How are monetary assets and liabilities valued
- at present value of future cash flows.
- **most assets and liabilities are monetary in nature
- MWe value most receivables and payables at the present value of future cash flows, reflecting an appropriate time value of money.
- Include money and claims to reveive money, the mamount of which is fixed or determinable.
- *cash and most receivables.
are obligations to pay amounts of cash, the amount of which is fixed or determinable. Most liabilities are monetary.
Show the time value of money. It is important to realize that the amount also equal the present value of future cash flows.
No Explicit Interest
Even though the agreement states a noninterest-bearing note, the amount does in fact include interest for the period of the loan.
FASB Statement of Financial Accounting Concepts No. 7 "Using cash flow information and present value in Accounting measurements"
- The statement provides a framework for using future cash flows as the basis for accounting measurement and assets that the objective in valuing an asset or liability using present alue is to approximate the fair value of that asset or liability.
- Key to that objective is determining the present value of future cash flows associated with the asset or liability, taking into account any uncertainity concerning the amounts and timeing of the cash flows.
- Leases are certain
- Law suits are uncertain (in present value calculations has been by discounting the "best estimate" of future cash flows applying a discount rate that has been adjusted to reflect the uncertainity of risk of those cash flows.
- The uncertainty is applied to the cash flows, not the discount rate.
- Probability is taken into account and the liability is only on the present value and not the interest associated with the uncertainty.
*The company's credit adjusted risk-free rate of interest is used when applying the expected cash flow approach to the calculation of present value.
the same amount is to be received or paid each period, the series of cash flows at the end of each period.
The first cash flow (receipt or payment) of an ordinary annuity is made one compounding period after the date on which the agreement begins.
Example: loan/leases (periodic interest is paidin equal amounts.) bonds (pay interest semiannually in the amount determined by multiplying a stated rate by a fixed principle amount)
- Referred to as an annuity in advance. Cash flows at the beginning of the period. First payment is made on the first day of the agreement.
- Example: lease for a rental of a building would be paid on the first day of the agreement.
Future value Ordinary Annuity
- In the FV of an ordinary annuity, the last cash payment will not earn any interest.
- FVA = 10000(annuity amount) x 3.31 = 33000
Future Value Annuity Due
In the future value of an annuity due, the last cash payment will earn interest.
- FVAD = 10000(annuity amount) x 3641 = 36410
- **unequal payments can't be solved with the chart of the annuity tables. The FV would have to be calculated seperately.
Present Value Annuity Due
- In the PV of an annuity due, no interest needs to be removed from the first cash payment. The second payment has one compounding period. The third period has two compunding periods that need to be removed.
- PVAD = 10000(annuity amount) x 2.73554 = 27355
Present Value Deferred Annuity
exsists when the first cash flow occurs more than one period after the date the agreement begins.
- **the pv of the deferred annuity can be calculated by summing the pv's of the three indivual cash flows, each discounted to todays PV.
- 2 step process:
- 1. Calculate the PV of the annuity as of the beginning of the annuity period.
- 2. Discount the single amount calculated in (1) to its present value as of today.
Solving for unknown values in Present Value Situations
- Four variables involving annuities:
- 1. Present Value of an ordinary annuity (PVA) or present value of an annuity dued (PVAD_
- 2. the amount of each annuity payment
- 3. the number of periods, n
- 4. the interest rate, i
valuation of long-term bonds
- The stated payments are equal to the contractual state rate multiplied be the face value of the bonds.
- At the date the bonds are issued (sold), the marketplace will detemine the price of the bonds based on the market rate of interest for investments with similar characteristics.
- The market rate at date of issuance may not equal the bonds' state rate in which cashe the price of the bonds (the amount the issuing company actually is borrowing) will not equal the bonds' face value.
- Bonds issued at more than face value are said to be issued with a premium.
- Bonds issued at less than face value are said to be issued at a discount.