Are the following pairs of events disjoint or not disjoint?
a. Arriving early for your statistics exam.
Arriving late for your statistics exam.
b. Drawing an ace from a standard deck of cards.
Drawing a diamond from a standard deck of cards.
The correct answer and explanation is :
a. Arriving early for your statistics exam vs. arriving late for your statistics exam.
These two events are disjoint.
Explanation:
In probability theory, two events are considered disjoint (or mutually exclusive) if they cannot happen at the same time. In this scenario, “arriving early for your statistics exam” means arriving before the exam start time, while “arriving late for your statistics exam” means arriving after the exam has started. Since it is not possible to arrive both early and late simultaneously, these two events cannot occur together. Therefore, they are disjoint.
Mathematically, the probability of both events happening at the same time is zero, i.e., ( P(\text{Arriving early and late}) = 0 ). Hence, the events are disjoint.
b. Drawing an ace from a standard deck of cards vs. drawing a diamond from a standard deck of cards.
These two events are not disjoint.
Explanation:
A standard deck of cards consists of 52 cards, including 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, and one of these 13 cards in the diamond suit is the ace of diamonds.
In this case, “drawing an ace from a standard deck of cards” refers to the event of selecting any one of the four aces (ace of hearts, ace of diamonds, ace of clubs, or ace of spades). “Drawing a diamond from a standard deck of cards” refers to the event of selecting any card from the diamond suit, which consists of 13 cards, including the ace of diamonds.
Since the ace of diamonds is included in both events (it is an ace and it is a diamond), it is possible for both events to occur simultaneously if you draw the ace of diamonds. Therefore, these two events are not disjoint because they share a common outcome — drawing the ace of diamonds.
Mathematically, the probability of both events happening at the same time (i.e., drawing the ace of diamonds) is non-zero, i.e., ( P(\text{Ace and Diamond}) > 0 ). Hence, the events are not disjoint.
In summary:
- a) Disjoint
- b) Not disjoint