Let's explore another way that two probabilities can be combined using the word or. First, let's find a simple probability. What is P of two? The probability of meeting the goal of drawing a two. We have four cards in total. Looking at the cards, we can see that two of them have the number two.
So, the number of favorable outcomes is two out of four total possible outcomes.
That gives us a probability of 2 over 4, which simplifies to 1/2.
Now let's look at an or probability.
What is P of two or square? This is the probability of meeting at least one of these two goals. The cards that are a two are the two circle and the two square. The cards that are a square are the one square and the two square. To find the outcomes that are a two or a square, we list all the unique cards that fit either description. That's the one circle, the two circle, and the two square. The two square meets both conditions, but we only count it once to avoid double counting. So, there are three favorable outcomes out of four total cards. That gives us a probability of 3/4s.
P of A or B is the probability that at least one of the events A or B happens.
Let's try another one with a new set of cards. What is the probability of drawing a card that is odd or has a star? This time we have five cards in total. The odd-numbered cards are the one square, the one star, and the three star. The cards with a star are the one star, the three star, and the four star.
Notice the one star and three star cards are in both groups. To find the total number of cards that are odd or a star, we count each unique card that fits the one square, the one star, the three star, and the four star. That gives us four unique cards out of five total. So, the probability is four fths. We can visualize this same idea using a vin diagram. To show the region that corresponds to P of odd or star, we need to select all the outcomes in the odd circle and all the outcomes in the star circle. This includes the part of the odd circle that doesn't overlap, the part of the star circle that doesn't overlap, and the overlapping section in the middle. Together, these three regions represent the union of the two events.
So, how do the values of P of A and P of A or B compare? Looking at the diagram, the region for A or B is the total shaded area of both circles. This area contains the entire circle for A plus any extra part of circle B. This means the probability of A or B must be greater than or equal to the probability of A. It can never be smaller. This is a fundamental property. P of A or B can never be smaller than the individual probabilities P of A and P of B because the event A or B includes all the outcomes of A and all the outcomes of B.
