200 people attended a blood bank and their blood types are recorded as follows: 50 have type A,65 have type B, 70 have type O and 15 have type AB.(a) Calculate the relative probability of each blood type.(b) If two people are selected without replacement, what is the probability that:(i) One person is type A and the other is type B.(ii) They both have type B blood.(iii) They both have the same type blood
Question
200 people attended a blood bank and their blood types are recorded as follows: 50 have type A,65 have type B, 70 have type O and 15 have type AB.(a) Calculate the relative probability of each blood type.(b) If two people are selected without replacement, what is the probability that:(i) One person is type A and the other is type B.(ii) They both have type B blood.(iii) They both have the same type blood
Solution
(a) The relative probability of each blood type is calculated by dividing the number of people with a specific blood type by the total number of people.
- The probability of type A is 50/200 = 0.25 or 25%
- The probability of type B is 65/200 = 0.325 or 32.5%
- The probability of type O is 70/200 = 0.35 or 35%
- The probability of type AB is 15/200 = 0.075 or 7.5%
(b) If two people are selected without replacement:
(i) The probability that one person is type A and the other is type B is calculated by multiplying the probability of selecting one person with type A by the probability of selecting one person with type B from the remaining people. This is (50/200) * (65/199) = 0.0812 or 8.12%.
(ii) The probability that they both have type B blood is calculated by multiplying the probability of selecting one person with type B by the probability of selecting another person with type B from the remaining people. This is (65/200) * (64/199) = 0.1053 or 10.53%.
(iii) The probability that they both have the same type blood is calculated by adding the probabilities of each blood type. This is:
- Type A: (50/200) * (49/199) = 0.0618 or 6.18%
- Type B: (65/200) * (64/199) = 0.1053 or 10.53%
- Type O: (70/200) * (69/199) = 0.1218 or 12.18%
- Type AB: (15/200) * (14/199) = 0.0105 or 1.05%
Adding these probabilities together gives 0.0618 + 0.1053 + 0.1218 + 0.0105 = 0.2994 or 29.94%.
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