Question 8

Scenario: The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group-blood group combinations.

Ethnic Groups O Blood A Blood B Blood AB Blood
1 0.082 0.120 0.011 0.004
2 0.121 0.141 0.018 0.003
3 0.215 0.206 0.059 0.020

Events:

  • - Type A selected.
  • - Type B selected.
  • - Ethnic group 3 selected.

(a)

Problem: Calculate , , and .

Process: The probability represents the total probability of picking someone with type A blood across all three ethnic groups. The probability represents the total probability of picking someone from ethnic group 3 across all four blood types. Both can be found by adding together the column for type A blood and the row for ethnic group 3 respectively. To find the probability of the intersection of the events in and , we can look at the table for the probability of people from ethnic group 3 with type A blood and find the value in the cell where they intersect.

Answer: 0.467, 0.5, and 0.206.

Summary

In a table like the one provided, where two types of characteristics are used as the axes, the summation of a row or column is the total probability for that specific characteristic and each cell represents the probability of the intersection (and) of the characteristic in that cell's row and column.

(b)

Problem: Calculate both and . Explain in context what each of these probabilities represents.

Process: To find the conditional probability , we must divide the probability of the intersection (and) of the events in and , , by the probability . To find the reverse of , , we must multiply the conditional probability by the probability ) and then divide the product by the probability .

Answer: 0.412 and 0.441. If a person has type A blood, the probability that he is from ethnic group 3 is given by . If we know that the individual came from ethnic group 3, the probability that he has type A blood is given by .

Summary

The conditional probability of given is:

The reverse of a conditional probability given is:

A conditional probability is read as "the probability of given ." It represents the probability that is true given than is also true.

(c)

Problem: If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1?

Process: The question can be rephrased to ask what the probability is that the selected individual is from ethnic group 1 given that they don't have type B blood. This can be represented with the conditional probability , where is the probability that an individual is from ethnic group 1 and is the probability that an individual doesn't have type B blood. The probability that an individual doesn't have type B blood is equal to the sum of all other probabilities in the table except for the ones in the column for type B blood.

Answer: 0.226.