Let's use the formal language of probability to be more precise. We have four cards. Two have circles and two have stars. The numbers are 1 2 3 and 4.
First, what's the probability of drawing a card with a circle? In other words, what is P of circle? This is written as P open parenthesy circle close parenthesy.
Two cards have circles. The probability is 2 out of 4 which simplifies to 1/2.
Now what if we want the probability of drawing a card that is a circle and has an even number? In formal probability language we ask what is P of circle and even. The word and means both conditions must be true. Card one is a circle but not even. Card two is a circle and even.
Card three is not a circle. Card four is even but not a circle. Only card two meets both conditions. With one favorable outcome out of four possible cards, the probability is 1/4. This brings us to formal notation. P of A is the probability that event A happens. P of A and B is the probability that both events A and B happen. In our example, event A was drawing a circle. Event B was drawing an even number. We can visualize this using a vin diagram.
Let's represent all circle outcomes with one circle and all even outcomes with another. The area for P of circle includes everything inside the circle event. That means cards that are circles and odd and cards that are circles and even.
What about P of circle and even? This corresponds to outcomes in both sets on the diagram. That's the overlapping section called the intersection. This region represents outcomes that satisfy the first condition and the second condition.
How does P of A and B relate to P of A? The area for A and B, that is the intersection, is part of the total area for A. The intersection can't be larger than the entire circle A. It can be the same size. If event B happens every time event A happens but therefore the probability of both A and B happening must be less than or equal to the probability of A happening on its own.
Visualizing probabilities on a ven diagram shows us that the probability of two events happening can never be larger than the probability of each event on its own.
