Transcript/notes Addition rules for probability
When discussing the 2 addition rules for probability, we first need to start with defining mutually exclusive events. And mutually exclusive events, sometimes called disjoint events, are events that cannot occur at the same time.
Let’s look at this from a visual perspective. Here is a venn diagram and the red circle represents probability of event A, and the green circle represents the probability of event B, and the rest of the area, the blue color in the square represents any other probabilities in the sample space. We know the total for probability is equal to one, so the total area in our diagram is equal to 1.
In our diagram, A and B have no overlapping areas, meaning, they cannot occur at the same time, so they are mutually exclusive. For instance probability of A could represent rolling a 3 on a die, probability of B could represent rolling a 6 on a die, and the rest of the blue area would represent rolling a 1, 2, 4, or 5.
Here is another diagram, but, probability of event A and probability of event B have an overlapping area, so these are not mutually exclusive, so they can occur at the same time. For instance let’s say you are picking a movie to watch from your personal collection. Probability of A could represent a movie that won best actress award, probability of B could represent a movie that won best actor award, the overlapping area of probability of A and B would represent a movie that won both best actress and best actor awards, and the rest of the area would then be a movie that did not win either of these awards.
Now to addition rules. The first addition rule of probability is that when 2 events, such as A and B are mutually exclusive, the probability that event A or event B will occur is, Probability of A or B equals probability of A plus probability of B. For instance, going back to our die roll example, where event A is rolling a 3 and event B is rolling a 6, so, again, they are mutually exclusive, we would use this rule. So, probability of A or B equals, 1 over 6, the probability of A, plus 1 over 6, the probability of b. And this ends up being 1 over 3, or 0.333 or 33.3%
This rule can extend beyond 2 events. Let’s say you are in a raffle with a total of 92 participants who have one raffle ticket each, and there will be 8 $100 gift cards awarded, 15 free oil changes awarded and 18 tee shirts awarded, with no repeat winners, so what is the probability that you win something?
I will also fill in this diagram as we go through the math. We can assign the $100 gift cards as event A, the oil changes as event B, and the tee shirts as event C, and since there are no repeat winners, these events are mutually exclusive. Our equation is probability of A or B or C equals probability of A plus probability of B plus probability of C.
With 92 participants, the probability of A equals 8 over 92, probability of B equals 15 over 92 and the probability of C equals 18 over 92. Add these up and we get 41 over 92, which equals 0.445 or a 44.5% of winning something.
Now to the second addition rule, which applies when 2 events, such as A and B are not mutually exclusive, so they can occur at the same time. And it is probability of A or B equals the probability of A plus the probability of B minus the probability of A and B.
Going back to our actress and actor award diagram, if we assign movies that won best actress as event A, and it has a probability of 0.18, and we assign movies that won best actor award as event B and it has a probability of 0.11. And the overlapping area of A and B has a probability of 0.03. Using our equation, we have the probability of A or B equals 0.18 plus 0.11 minus 0.03, which is 0.26, or a 26% of a movie that won best actress or best actor.
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