What Is the Addition Rule for Probabilities?

The addition rule for probabilities describes two formulas, one for the probability for either of two mutually exclusive events happening and the other for the probability of two non-mutually exclusive events happening.

The first formula is just the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur.

The Formulas for the Addition Rules for Probabilities Is

Mathematically, the probability of two mutually exclusive events is denoted by:


$P(Y \text{ or } Z) = P(Y)+P(Z)$P(Y or Z)=P(Y)+P(Z)

Mathematically, the probability of two non-mutually exclusive events is denoted by:


$P(Y \text{ or } Z) = P(Y) + P(Z) – P(Y \text{ and } Z)$P(Y or Z)=P(Y)+P(Z)−P(Y and Z)

What Does the Addition Rule for Probabilities Tell You?

To illustrate the first rule in the addition rule for probabilities, consider a die with six sides and the chances of rolling either a 3 or a 6. Since the chances of rolling a 3 are 1 in 6 and the chances of rolling a 6 are also 1 in 6, the chance of rolling either a 3 or a 6 is:

1/6 + 1/6 = 2/6 = 1/3

To illustrate the second rule, consider a class in which there are 9 boys and 11 girls. At the end of the term, 5 girls and 4 boys receive a grade of B. If a student is selected by chance, what are the odds that the student will be either a girl or a B student? Since the chances of selecting a girl are 11 in 20, the chances of selecting a B student are 9 in 20 and the chances of selecting a girl who is a B student are 5/20, the chances of picking a girl or a B student are:

11/20 + 9/20 – 5/20 =15/20 = 3/4

In reality, the two rules simplify to just one rule, the second one. That’s because in the first case, the probability of two mutually exclusive events both happening is 0. In the example with the die, it’s impossible to roll both a 3 and a 6 on one roll of a single die. So the two events are mutually exclusive.

Mutual Exclusivity

Mutually exclusive is a statistical term describing two or more events that cannot coincide. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other.  For a basic example, consider the rolling of dice. You cannot roll both a five and a three simultaneously on a single die. Furthermore, getting a three on an initial roll has no impact on whether or not a subsequent roll yields a five. All rolls of a die are independent events.

What Is the Advance/Decline (A/D) Line?

The advance/decline line (or A/D line) is a technical indicator that plots the difference between the number of advancing and declining stocks on a daily basis. The indicator is cumulative, with a positive number being added to the prior number, or if the number is negative it is subtracted from the prior number.

The A/D line is used to show market sentiment, as it tells traders whether there are more stocks rising or falling. It is used to confirm price trends in major indexes, and can also warn of reversals when divergence occurs.

The Formula for Advance/Decline (A/D) Line Is:


$\begin{aligned} &\text{A/D} = \text{Net Advances} + \begin{cases} \text{PA, if PA value exists} \\ \text{0, if no PA value} \\ \end{cases} \\ &\textbf{where:} \\ &\text{Net Advances} = \text{Difference between number of daily} \\ &\text{ascending and declining stocks} \\ &\text{PA} = \text{Previous Advances} \\ &\text{Previous Advances} = \text{Prior indicator reading} \\ \end{aligned}$​A/D=Net Advances+{PA, if PA value exists0, if no PA value​where:Net Advances=Difference between number of dailyascending and declining stocksPA=Previous AdvancesPrevious Advances=Prior indicator reading​

How to Calculate the A/D Line

1. Subtract the number of stocks that finished lower on the day from the number of stocks that finished higher on the day. This will give you the Net Advances.
2. If this is the first time calculating the average, the Net Advances will be the first value used for the indicator.
3. On the next day, calculate the Net Advances for that day. Add to the total from the prior day if positive or subtract if negative.
4. Repeat steps one and three daily.

What Does the A/D Line Tell You?

The A/D line is used to confirm the strength of a current trend and its likelihood of reversing. The indicator shows if the majority of stocks are participating in the direction of the market. 

If the indexes are moving up but the A/D line is sloping downwards, called bearish divergence, it’s a sign that the markets are losing their breadth and may be about to reverse direction. If the slope of the A/D line is up and the market is trending upward, then the market is said to be healthy.

Conversely, if the indexes are continuing to move lower and the A/D line has turned upwards, called bullish divergence, it may be an indication that the sellers are losing their conviction. If the A/D line and the markets are both trending lower together, there is a greater chance that declining prices will continue.

Difference Between the A/D Line and Arms Index (TRIN)

The A/D line is typically used as a longer-term indicator, showing how many stocks are rising and falling over time. The Arms Index (TRIN), on the other hand, is typically a shorter-term indicator that measures the ratio of advancing stocks to the ratio of advancing volume. Because the calculations and the time frame they focus on are different, both these indicators tell traders different pieces of information.

Limitations of Using the A/D Line

The A/D line won’t always provide accurate readings in regards to NASDAQ stocks. This is because the NASDAQ frequently lists small speculative companies, many of which eventually fail or get delisted. While the stocks get delisted on the exchange, they remain in the prior calculated values of the A/D line. This then affects future calculations which are added to the cumulative prior value. Because of this, the A/D line will sometimes fall for extended periods of time, even while NASDAQ-related indexes are rising.

Another thing to be aware of is that some indexes are market capitalization weighted. This means that the bigger the company the more impact they have on the index’s movement. The A/D line gives equal weight to all stocks. Therefore, it is a better gauge of the average small to mid-cap stock, and not the fewer in number large or mega-cap stocks.

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What Is an 8-K?

An 8-K is a report of unscheduled material events or corporate changes at a company that could be of importance to the shareholders or the Securities and Exchange Commission (SEC). Also known as a Form 8K, the report notifies the public of events, including acquisitions, bankruptcy, the resignation of directors, or changes in the fiscal year.

Understanding Form 8-K

An 8-K is required to announce significant events relevant to shareholders. Companies usually have four business days to file an 8-K for most specified items.

Documents fulfilling Regulation Fair Disclosure (Reg FD) requirements may be due before four business days have passed. An organization must determine if the information is material and submit the report to the SEC. The SEC makes the reports available through the Electronic Data Gathering, Analysis, and Retrieval (EDGAR) platform.

The SEC outlines the various situations that require Form 8-K. There are nine sections within the Investor Bulletin. Each of these sections may have anywhere from one to eight subsections. The most recent permanent change to Form 8-K disclosure rules occurred in 2004.

Benefits of Form 8-K

First and foremost, Form 8-K provides investors with timely notification of significant changes at listed companies. Many of these changes are defined explicitly by the SEC. In contrast, others are simply events that firms consider to be sufficiently noteworthy. In any case, the form provides a way for firms to communicate directly with investors. The information provided is not filtered or altered by media organizations in any way. Furthermore, investors do not have to watch TV programs, subscribe to magazines, or even wade through financial news websites to get the 8-K.

Form 8-K also provides substantial benefits to listed companies. By filing an 8-K in a timely fashion, the firm’s management can meet specific disclosure requirements and avoid insider trading allegations. Companies may also use Form 8-K to notify investors of any events that they consider to be important.

Finally, Form 8-K provides a valuable record for economic researchers. For example, academics might wonder what influence various events have on stock prices. It is possible to estimate the impact of these events using regressions, but researchers need reliable data. Because 8-K disclosures are legally required, they provide a complete record and prevent sample selection bias.

Criticism of Form 8-K

Like any legally required paperwork, Form 8-K imposes costs on businesses. There is the cost of preparing and submitting the forms, as well as possible penalties for failing to file on time. Although it is only one small part of the problem, the need to file Form 8-K also deters small companies from going public in the first place. Requiring companies to provide information helps investors make better choices. However, it can reduce their investment options when the burden on businesses becomes too high.

Requirements for Form 8-K

The SEC requires disclosure for numerous changes relating to a registrant’s business and operations. Changes to a material definitive agreement or the bankruptcy of an entity must be reported. Other financial information disclosure requirements include the completion of an acquisition, changes in the firm’s financial condition, disposal activities, and substantial impairments. The SEC mandates filing an 8-K for the delisting of a stock, failure to meet listing standards, unregistered sales of securities, and material modifications to shareholder rights.

An 8-K is required when a business changes accounting firms used for certification. Changes in corporate governance, such as control of the registrant or amendments to articles of incorporation, need to be reported. Changes in the fiscal year and modifications of the registrant’s code of ethics must also be disclosed.

The SEC also requires a report upon the election, appointment, or departure of a director or specific officers. Form 8-K must be used to report changes related to asset-backed securities. The form may also be used to meet Regulation Fair Disclosure requirements.

Form 8-K reports may be issued based on other events up to the company’s discretion that the registrant considers to be of importance to shareholders.