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Addition Rule of Probability
There are many rules associated with solving probability problems. This lesson deals with the addition rule. The addition rule helps you solve probability problems that involve two events. Even though we discuss two events (usually labeled A and B), we’re really talking about performing one task (rolling dice, drawing cards, spinning a spinner, etc.) and finding about the probability of two things happening in that one task.
 
When asked to find the probability of A or B, we mean that A can happen, or B can happen, or both can happen together. This is what is stated in the addition rule.
 
The Addition Rule:
 
Consider events A and B. P(AB)= P(A) + P(B) - P(AB)
 
What The Rule Means:
 
Suppose we roll two dice and want to find the probability of rolling a sum of 6 or 8. This can be written in words as P(6 or 8) or more mathematically is P(68). Remember that OR (the union symbol ) means that one or the other or both events can happen. So what is the probability of getting a 6 or an 8 or both? You may want to refer to the dice chart in introductory lesson on basic probability if you need to familiarize yourself with the outcomes of rolling two dice.
 
P(6) = 5/36
P(8) = 5/36
P(6 and 8 together) is impossible so the probability is 0.
 
So P(68) = 5/36 + 5/36 - 0 = 10/36 = 5/18
 
Since rolling a sum of 6 and 8 cannot happen together at the same time, we say that they are disjoint or mutually exclusive. When two events are disjoint, you do not have to worry about subtracting the probability of both events happening together since that probability will always be 0. Since it can sometimes save a step, a lot of students like to determine if events are disjoint before finding individual probabilities
 
Let's Practice:
 
  1. You are going to pull one card out of a deck. Find P(AceKing).
 
The addition rule says we need to find P(Ace) + P(King) - P(both). If you need to familiarize yourself with the features of a deck of cards, refer to introductory lesson on basic probability for more information.
 
P(Ace) = 4/52
P(King) = 4/52
P(both at the same time) = 0
 
P(AceKing) = 4/52 + 4/52 = 8/52 = 2/13
 
Notice that the final answer is always simplified. However, most of the time it is best to NOT simplify along the way so that you’ll have easy common denominators.
 
  1. You are going to roll two dice. Find P(sum that is even or sum that is a multiple of 3).
 
The addition rule says we need to find P(even) + P(multiple of 3) - P(both).

If you need to familiarize yourself with the possible combinations of rolling two dice, refer to introductory lesson on basic probability for a chart showing all 36 possibilities.
 
P(even) means how many ways to roll 2, 4, 6, 8, 10, or 12.
P(even) = 18/36
 
P(multiple of 3) means how many ways to roll 3, 6, 9 or 12.
P(multiple of 3) = 12/36
 
P(both) means what is the overlap. Notice that 6 and 12 occur in both places and have been counted twice. We need to subtract those out.
P(both) = 6/36
 
So P(sum that is even or a multiple of 3) = 18/36 + 12/36 - 6/36 = 24/36 = 2/3.

Example Group #1
The first three problems involve drawing a card from an ordinary deck of cards.
Example P(twored)
What is your answer?
 
Example P(three or jack)
What is your answer?
 
Example P(club or four)
What is your answer?
 

Example Group #2
The last two problems involve rolling two dice.
Example P(sum that is odd  sum that is even)
What is your answer?
 
Example P(sum of 4 or sum that is a multiple of 3)
What is your answer?
 



S Taylor

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