In both cases, we find the individual probabilities, P(A) or P(B), by dividing the favorable outcomes by the total number of outcomes.
Disjoint Events
We can see that the probabilities of disjoint events do not share any region in the diagram. This means that there is no favorable outcome that satisfies both A and B. Therefore, is impossible for A and B to occur at the same time. Let's write this as a probability.
P(AandB) = 0
Similarly, since disjoint events do not share any outcomes, to find the probability that A or B occurs, we can add their probabilities.
P(AorB) = P(A) + P(B)
Overlapping Events
We can see that overlapping events share a region in the diagram. This is because there is at least one possible outcome that is favorable for both A and B. Therefore, the probability that A and B occur is always greater than 0.
P(AandB) > 0
Looking at the diagram, we can see that if we only added the probabilities of A and B, we would add the probability P(AandB) twice. Because of this, we need to subtract this probability from the sum to find the probability that A or B occurs.
P(AorB) = P(A) + P(B) - P(AandB)
Example
Let's consider some possible events that could occur when using a standard deck of cards.
Disjoint Events Example
First, let's say that we want to know the probability that we will randomly draw a 2 or a 10. Since there are no favorable outcomes that would satisfy both events, these are disjoint events. Let's draw these events with a Venn diagram!
Since there are 52 cards in a deck and 4 types of cards, we will divide 4 by 52 to find the probability of each individual event.
ccc
Drawing a10: && Drawing a2: [0.5em]
P(10) = 4/52 && P(2) = 4/52
We know that there are no favorable outcomes for both events. Therefore, the probability of drawing a card with a 2 and a 10 is 0.
P(10 and 2) = 0
Now we will find the probability that we draw either a 10 or a 2. Since the events are disjoint, we will find this probability by adding the individual probabilities.
We can confirm this probability seeing that 8 of the 52 cards are favorable outcomes.
Overlapping Events Example
Now consider the events of drawing a 10 and a spade.
The probability of drawing a 10 is the same as in the previous example, 452. Since there are 13 of each type of card, we find the probability of drawing a spade by dividing 13 by 52.
P(spade) = 13/52
We can see that there is a favorable outcome that satisfies both events: drawing the 10 of spades. Let's find the probability of drawing a card that is a 10 and a spade.
P(10 and spade) = 1/52
Now let's find the probability of drawing a 10 or drawing a spade.
P(10 or spade) = P(10) + P(spade) - P(10 and spade)
