# The Addition Rule and Complements Page 1. Blood Types. The purpose of this activity is to introduce you to the addition rules of probability.

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1 The Addition Rule and Complements Page 1 Blood Types The purpose of this activity is to introduce you to the addition rules of probability. The addition rules of probability are used to find the probabilities of compound events (an event involving more than one outcome). Specifically, it involves finding the probability of either one of two or more event occurring. Blood Typing According to Craig Medical online (www.craigmedical.com), blood typing is determined by the type of antigens or markers that are on the surface of red blood cells (either A or B ) and if there are antibodies to a portion of the blood type known as the Rh factor (either positive or negative ). There are four blood types: A, B, AB, and O and two Rh factors: positive and negative. The statistical percentage of type and Rh factor in human blood for the general population is given below: Blood Type % Frequency O 45% A 40% B 11% AB 4% Rh factor % Frequency positive 84% negative 16% 1. Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no outcome in common). Examples Selecting a male and selecting a female. Selecting a striped pool ball and selecting a solid pool ball. Selecting a Honda and selecting a Saturn. When considering the blood type of a person, provide two events that are mutually exclusive involving (a) blood type (b) Rh factor.

2 The Addition Rule and Complements Page 2 Consider a sample of 360 people who represent the general population s blood typing: Blood Type # of people O 162 A 144 B 39 AB 15 Rh factor # of people positive 302 negative When two events are mutually exclusive, there is no possibility of overlap in the events occurring. The Venn diagram below presents a pictorial representation of the events of selecting a person with Type A blood and selecting a person with Type B blood. Type A Type B Other Types (AB or O) 177 a. What is the frequency of people with Type A blood? b. What is the frequency of people with Type B blood? c. What is the frequency of people with either Type A or Type B blood? Explain how you got this answer. d. What is the probability that a person randomly selected from this sample has either Type A or Type B blood?

3 The Addition Rule and Complements Page 3 Addition Rule 1 When two events A and B are mutually exclusive, the probability that A or B will occur is P(A or B) = P(A) + P(B) Addition Rule 1 simply states that whenever we have two events that cannot occur at the same time, then the probability of either one or the other occurring is found by adding their individual probabilities together. 3. For the perfect sample above, what is the probability that a person randomly selected from the sample has either Type A or Type AB blood? According to Craig Medical online (www.craigmedical.com), the overall statistical distribution of blood type plus Rh factor in the general population is as follows: Rh factor Total Blood Type + O 38% 7% 45% A 34% 6% 40% B 9% 2% 11% AB 3% 1% 4% Total 84% 16% 100% 4. Are the two events of selecting a person with Type O blood and selecting a person with Rh factor negative mutually exclusive? Explain. 5. Determine the following probabilities when randomly selecting a person from the general population: a. P(Type A) = b. P(Rh positive) = c. P(Type A and Rh positive) =

4 The Addition Rule and Complements Page 4 Consider the same sample of 360 people who represent the general population s blood typing: Rh factor Total Blood Type + O A B AB Total When two events are not mutually exclusive, there is an overlap in the events occurring. That is, both events can occur at the same time. The Venn diagram below presents a pictorial representation of the events of selecting a person with Type O blood and selecting a person with an Rh positive factor. Type O Rh positive Other Types 33 a. What is the frequency of people with Type O blood? Shade the region above with the following lines: b. What is the frequency of people with Rh positive blood? Shade the region above with the following lines: c. What is the frequency of people with both Type O and Rh positive blood? How is it shades in the diagram above?

5 The Addition Rule and Complements Page 5 Notice if we were to determine the probability of randomly selecting a person with Type O or Rh positive blood and tried to add the probabilities together (as in Rule 1), we end up over counting the people because 38 people have both Type O and Rh positive blood. That is, applying Rule 1 we get P(Type O) + P(Rh positive) = + = But this can t be since no probability can be over 1. However, if we subtract the people we double counted in the center (who have both Type O and Rh positive blood) we get P(Type O) + P(Rh positive) P(Type O and Rh positive) = + = Addition Rule 2 If A and B are not mutually exclusive, the probability that A or B will occur is P(A or B) = P(A) + P(B) P(A and B) Addition Rule 2 simply states that whenever we have two events that can occur at the same time, then the probability of either one or the other occurring is found by adding their individual probabilities together and then subtracting the probability of both occurring at the same time. The reason for subtracting P(A and B) is because the individual probabilities double counted the probability that both occurred at the same time. 7. For the perfect sample above, find the following probabilities when a person randomly selected a. The probability that the person has either Type B or Rh negative blood b. P(Type AB or Rh positive) =

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