This is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other. Two random variables are independent if the realization of one does not affect the probability distribution of the other; when dealing with collections of more than two events, a weak and a strong notion of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while saying that the events are mutually independent intuitively means that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables; the name "mutual independence" seems the outcome of a pedagogical choice to distinguish the stronger notion from "pairwise independence", a weaker notion.

In the advanced literature of probability theory and stochastic processes, the stronger notion is named independence with no modifier. It is stronger. Two events A and B are independent if and only if their joint probability equals the product of their probabilities: Why this defines independence is made clear by rewriting with conditional probabilities: P = P P ⟺ P = P P = P.and P = P P ⟺ P = P. Thus, the occurrence of B does not affect the probability of A, vice versa. Although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if P or P are 0. Furthermore, the preferred definition makes clear by symmetry that when A is independent of B, B is independent of A. Stated in terms of log probability, two events are independent if and only if the log probability of the joint event is the sum of the log probability of the individual events: log ⁡ P = log ⁡ P + log ⁡ P In information theory, negative log probability is interpreted as information content, thus two events are independent if and only if the information content of the combined event equals the sum of information content of the individual events: I = I + I See Information content § Additivity of independent events for details.

Stated in terms of odds, two events are independent if and only if the odds ratio of A and B is unity. Analogously with probability, this is equivalent to the conditional odds being equal to the unconditional odds: O = O and O = O, or to the odds of one event, given the other event, being the same as the odds of the event, given the other event not occurring: O = O and O = O; the odds ratio can be defined as O: O, {\displaystyle O(A\mid

Voices of Gotham is a competitive and performing chorus composed of 60 singers from New York City and surrounding areas. VoG represents the'Hell's Kitchen, New York' chapter of the Barbershop Harmony Society, is the 2013 Mid-Atlantic District Chorus Champion. VoG was most designated the 10th best barbershop chorus in the world in the 2017 Barbershop Harmony Society Chorus Competition in Las Vegas, Nevada. Voices of Gotham was founded by 12'charter members' on January 8, 2008 and holds weekly auditions for prospective singers. Voices of Gotham hosts an annual event called the Tin Pan Alley Novice Quartet Contest each year in varying locations in New York City. A large number of quartets, many assembled for the event, sing one song each in pursuit of their goal to "Win the Tin." Despite the somewhat informal nature of the event, a panel of accredited judges from the Barbershop Harmony Society adjudicate the event. In addition to standard Gold and Bronze medals, the winning quartet is awarded the coveted Tin Pan, an antique tin pan inscribed with the quartet names of all prior champions.

October 2011 – Mid-Atlantic District Chorus Champions July 2012 – First time International Barbershop Harmony Society Chorus Competitor. July 2013 – First time finishing in the top 10 in Barbershop Harmony Society International Chorus contest. February 2014 – Awardee of funding from the Lower Manhattan Cultural Council – Fund for Creative Communities in support of its 2014 Performance Series. August 2014 – Awardee of funding from the New York City Department of Cultural Affairs Cultural Development Fund in support of its 2014–2015 Performance Series. July 2015 – Awardee of funding from the Barbershop Harmony Society Innovation Grant Program in support of its 2015 Arrangers Lab Program. July 2017 – Top-ten finisher in the Barbershop Harmony Society's International Chorus Competition. Featured on WSRK 103.9 FM as promotion for guest appearance on Oneonta, NY chorus show following Hurricane Sandy. Participants in the 2012 Phoenicia International Festival of the Voice for "Barbershop Day." Listed as one of prominent a cappella groups in New York City as part of Time Out New York article.

"There’s no better way to establish your own a cappella group than to meet and pry away a few members of this versatile male barbershop chorus—the Voices of Gotham." Official website

2nd Battalion 11th Marines is an artillery battalion comprising three firing batteries and a Headquarters Battery. The battalion is stationed at Marine Corps Base Camp Pendleton and its primary weapon system is the M777 lightweight howitzer; the battalion was the second in the Marine Corps to transition from the M198 Howitzer. They fall under the command of the 1st Marine Division. Provide direct support of the 11th Marine Regiment in time of conflict; that support may come in the traditional fashion of artillery support to maneuver forces, or by providing batteries to serve as provisional rifle companies. They have the secondary mission of conducting civil-military operations or "CAG Ops". Headquarters Battery Battery E Battery F Battery G The battalion was activated September 5, 1918, at Montierchame, France as the 2d Battalion, 11th Regiment, they were relocated during July -- August 1919 to Virginia. And were deactivated August 11, 1919. 2/11 was reactivated on May 9, 1927, at Marine Corps Base Quantico, Virginia, as the 2d Battalion, 11th Regiment.

Deployed during May 1927 to Chinandega and assigned to the 2d Brigade. They participated in operations against dissidents in May–July 1927, they were redesignated July 1, 1927, as the 2d Separate Battalion, 11th Regiment and relocated to San Diego, California. The battalion was deactivated September 10, 1927, they were again reactivated on January 5, 1928, at San Diego, California, as the 2d Battalion, 11th Regiment. The battalion deployed during January 1928 to Corinto and assigned to the 2d Brigade, they were deactivated August 20, 1929, on board the USS Henderson while en route to the United States. 2/11 was again reactivated on January 1, 1941, at Guantanamo Bay, Cuba as the 2d Battalion, 11th Marines, 1st Brigade. Relocated during April 1941 to Parris Island, South Carolina, they deployed during June 1942 to New Zealand. The battalion participated in the following World War II battles: Battle of Guadalcanal, Battle of New Britain, Battle of Peleliu and the Battle of Okinawa, they redeployed during September 1945 to Tientsin and participated in the occupation of Northern China from September 1945 to December 1946.

They were reassigned during January 1947 to the Fleet Marine Force and redeployed during January - February 1947 to Guam. Relocated during April–May 1947 they moved to Marine Corps Base Camp Pendleton and were reassigned to the 3rd Marine Brigade, they were again reassigned during July 1947 to the 1st Marine Division and relocated during August 1947 to Marine Corps Logistics Base Barstow, California. 2/11 was deactivated on September 28, 1947. The battalion was reactivated on August 7, 1950, at Camp Pendleton as the 2nd Battalion, 11th Marines, 1st Marine Division and deployed during August–September 1950 to Kobe, Japan, they participated in the Korean War, operating at Inchon-Seoul, the Chosin Reservoir, East Central front, Western front. They were stationed at the Korean Demilitarized Zone from July 1953-March 1955 and relocated during March 1955 to Camp Pendleton, California. 2/11 deployed during February–March 1966 to Camp Hansen, Okinawa. They again redeployed during May 1966 to Republic of Vietnam and participated in the Vietnam War from March 1966 to March 1971, operating from Chu Lai, Tam Ky, Hue/Phu Bai, Thon Back Thack, Cao Doi and An Hoa.

They returned in March -- April 1971 to California. Deployed during December 1990 with Regimental Landing Team 5, 5th Marine Expeditionary Brigade to Southwest Asia, they participated in the defense of Saudi Arabia until January 1991 and proceeded to participate in the liberation of Kuwait in January - May 1991 operating in Um Guar Al-Juba, Al Wafrah. Following the war they took part in the Bangladesh relief effort Operation Sea Angel in May 1991; the battalion returned on June 29, 1991, to Camp Pendleton and was last the Gulf War combat unit to return stateside. 1998- Battery F was attached to 31st MEU BLT 2/4 Battery F Awarded the Armed Forces Expeditionary Medal for Combat Operations and Naval Unit Commendations, Meritirous Unit Commendation and Sea Service Deployment Ribbon. The flexibility of the MEU was demonstrated with the Iraq crisis in late 1998 regarding the regime not complying with the U. N. weapons inspections process. All four ARG ships had just completed Exercise Foal Eagle off the coast of Korea, were heading to various port visits for liberty, when each ship received the call in early Nov 1998 to sail to Okinawa to onload the 31st MEU.

A significant portion of the 31st MEU’s 2000 Marines were engaged in urban warfare training in Guam when their message to return to Okinawa came in November. The rest were still in Okinawa, but a quarter of those were a new infantry battalion, just rotating in from California; the battalion had just two days to gather all their personnel to get ready to deploy. The 31st MEU and ships’ company personnel started their initial onloads to the ships on 9 Nov and completed the morning of 11 Nov. In one night alone, they loaded more than 170 pallets of equipment and cargo. In addition, a C-5 Galaxy from Marine Corps Air Station El Toro scheduled to bring maintenance supplies and tools to Okinawa two weeks arrived early on 10 Nov 1998 in order to restock the MEU’s Air Combat Element; this evolution was a part of the normal supply rotation, but the shipment arrived a week early – just in time to load onto the ships before they departed. From Nov 1998 to Feb 1999, the MEU participated in operations in the Persian Gulf and off the coast of Kuwait, including Operation Southern Watch and Operation Des