Disjoint Events

Disjoint Events Mutually Exclusive Events Probability Statistics Probability Theory Event Intersection Probability Space Event Analysis
Disjoint events, also known as mutually exclusive events, are pairs of events in a probability space that cannot occur at the same time.
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Definition

Disjoint Events, also known as Mutually Exclusive Events, are pairs of events in a probability space that cannot occur simultaneously. In formal terms, two events \(A\) and \(B\) are disjoint if their intersection is empty, that is, \(A \cap B = \emptyset\). This denotes that if event \(A\) occurs, event \(B\) cannot occur, and vice versa.

Examples

  1. Tossing a Coin: Consider the events of getting a head (H) or a tail (T) in a single coin toss. The events are disjoint because you cannot get both a head and a tail in one toss.

    • Event A (Head): {H}
    • Event B (Tail): {T}
    • \(A \cap B = \emptyset\)

  2. Rolling a Die: When rolling a six-sided die, the events of getting an even number (2, 4, 6) and an odd number (1, 3, 5) are disjoint because you cannot roll an even and odd number simultaneously.

    • Event A (Even): {2, 4, 6}
    • Event B (Odd): {1, 3, 5}
    • \(A \cap B = \emptyset\)

  3. Drawing a Card: Choosing a card from a standard deck of 52 cards, the events “drawing a Spade” and “drawing a Heart” are disjoint because a card cannot be both a Spade and a Heart.

    • Event A (Spade): {all Spades}
    • Event B (Heart): {all Hearts}
    • \(A \cap B = \emptyset\)

Frequently Asked Questions

Q1. Can two events be independent and disjoint at the same time? No, two events cannot be independent and disjoint at the same time, except if one of them has a probability of zero. For disjoint events, the occurrence of one event excludes the occurrence of the other, making them dependent by definition.

Q2. How do you calculate the probability of either of two disjoint events occurring? For two disjoint events \(A\) and \(B\), the probability of either occurring is the sum of their individual probabilities: \[ P(A \cup B) = P(A) + P(B) \]

Q3. Are complementary events always disjoint? Yes, complementary events are always disjoint. The occurrence of an event and its complement are mutually exclusive, as the complement represents all outcomes not included in the event.

  • Probability Space: A mathematical construct that provides a formal model for random phenomena. It consists of a sample space, a set of events, and a probability measure.
  • Intersection (\(\cap\)): The set of outcomes that are common to two or more events.
  • Union (\(\cup\)): The set of all outcomes that are in at least one of the events.
  • Complement: For an event \(A\), the complement (\(A^c\)) consists of all outcomes in the sample space that are not in \(A\).

Online References

  1. Investopedia on Mutually Exclusive Events
  2. Wikipedia on Mutual Exclusivity
  3. Khan Academy - Probability: Mutually exclusive events

Suggested Books for Further Studies

  1. “Introduction to Probability” by Dimitri P. Bertsekas and John N. Tsitsiklis
  2. “A First Course in Probability” by Sheldon Ross
  3. “Probability and Statistics for Engineers and Scientists” by Ronald E. Walpole, Raymond H. Myers, and Sharon L. Myers
  4. “An Introduction to Probability Theory and Its Applications” by William Feller

Fundamentals of Disjoint Events: Statistics Basics Quiz

### What is another term for disjoint events? - [ ] Independent events - [ ] Complementary events - [x] Mutually exclusive events - [ ] Dependent events > **Explanation:** Disjoint events are also known as mutually exclusive events because the occurrence of one event excludes the occurrence of the other. ### Can two disjoint events happen at the same time? - [ ] Yes, they can happen simultaneously. - [x] No, they cannot happen simultaneously. - [ ] Only if they are independent. - [ ] Only if they are dependent. > **Explanation:** By definition, disjoint events cannot happen at the same time. If one event occurs, the other cannot. ### Two events \\(A\\) and \\(B\\) are disjoint. What is \\(P(A \cap B)\\)? - [ ] \\(P(A) \times P(B)\\) - [x] 0 - [ ] \\(P(A) + P(B)\\) - [ ] None of the above > **Explanation:** For disjoint events, the probability of both occurring simultaneously, \\(P(A \cap B)\\), is zero because their intersection is empty. ### If Event A is drawing an Ace from a deck of cards and Event B is drawing a King, are A and B disjoint? - [x] Yes, they are disjoint. - [ ] No, they are not disjoint. - [ ] Only if the deck is reshuffled. - [ ] Only if both cards are drawn. > **Explanation:** Drawing an Ace and drawing a King are disjoint because you cannot draw both at the same time in a single draw. ### What is the formula to find the probability of either of two disjoint events happening? - [x] \\(P(A \cup B) = P(A) + P(B)\\) - [ ] \\(P(A \cap B) = P(A) + P(B)\\) - [ ] \\(P(A) \times P(B)\\) - [ ] \\(P(A \cap B) = P(A) \times P(B)\\) > **Explanation:** For disjoint events, the probability of either event \\(A\\) or event \\(B\\) happening is the sum of their individual probabilities. ### What does the intersection of two disjoint events equal? - [ ] \\(\Omega\\) (the sample space) - [x] \\(\emptyset\\) (the empty set) - [ ] A non-zero set - [ ] The union of the events > **Explanation:** The intersection of two disjoint events is the empty set, \\(\emptyset\\), because they cannot both happen simultaneously. ### If \\(P(A) = 0.3\\) and \\(P(B) = 0.4\\) for two disjoint events, what is \\(P(A \cup B)\\)? - [x] 0.7 - [ ] 0.12 - [ ] 0.3 - [ ] 1 > **Explanation:** For disjoint events \\(A\\) and \\(B\\), \\(P(A \cup B) = P(A) + P(B) = 0.3 + 0.4 = 0.7\\). ### If two events are disjoint, can they be dependent? - [ ] Yes, always. - [x] No, they cannot be dependent. - [ ] It depends on the scenario. - [ ] Only when their probabilities are zero. > **Explanation:** Disjoint events are dependent because the occurrence of one event completely excludes the occurrence of the other. ### If rolling a die, are the events of rolling a 3 and rolling an odd number disjoint? - [ ] Yes, they are disjoint. - [x] No, they are not disjoint. - [ ] Only with a biased die. - [ ] Only if the die has repeated numbers. > **Explanation:** The event "rolling a 3" is part of the event "rolling an odd number," so they are not disjoint. ### Which of the following pairs represent disjoint events when drawing a card? - [ ] Drawing a 5 and drawing a red card - [x] Drawing a face card and drawing a numbered card - [ ] Drawing a heart and drawing a spade - [ ] Drawing a club and drawing a king > **Explanation:** "Drawing a face card" and "drawing a numbered card" are disjoint, as a card cannot be categorized as both a face card and a numbered card.

Thank you for exploring the concept of disjoint events with us. Happy studying!

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Wednesday, August 7, 2024
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