One of the most important concepts in probability theory is the concept of mutually exclusive. Two events are said to be mutually exclusive when the occurrence of one of the events implies non-occurrence of the other in the same trial (Gnedenko, 2017). This can be explained with the help of an example. Let an event be defined as drawing a ball from an urn containing blue, black and red balls. Now, the ball drawn from the urn will either be a red ball or a black ball or a blue ball. The ball drawn cannot be black and red at the same time or red and blue at the same time. Thus, the event is said to be mutually exclusive.
On the other hand, collectively exhaustive event is another most important concept in probability theory. A set of events will be termed as collectively exhaustive events if at least one of the events have occurred in a trial (Von Mises, 2014). Considering the same example of drawing a ball from the urn containing red, black or blue ball, any one of the three balls blue, black or red will be drawn. There are no other types of balls present in the urn that can be drawn other than these three colors. Thus, drawing a ball from the urn is also a collectively exhaustive event.
Considering another example of a deck of 52 cards. Let an event be defined as drawing a king or a queen from the deck of 52 cards. This even so defined is mutually exclusive as while drawing a card, a king and a queen cannot appear together. It will either be a king or a queen. On the other hand, while drawing a card from the deck, cards other than the king or queen can also be drawn as king and queen are not the only options in the deck. Thus, the event is not collectively exhaustive. Hence, it can be said that all mutually exclusive events are not collectively exhaustive.
Gnedenko, B. V. (2017). Theory of probability. Routledge.
Von Mises, R. (2014). Mathematical theory of probability and statistics. Academic Press.