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Blood phenotypes refer to the different blood types found within a human population. These types, known as A, B, AB, and O, result from specific antigens present on red blood cells. Each phenotype has its own frequency in any given population, which can change based on genetic diversity and other factors. In our example:

• Phenotype A occurs with a probability of 0.42
• Phenotype B with 0.10
• Phenotype AB with 0.04
• Phenotype O with 0.44

The sum of all these probabilities must equal 1, as these are all the potential outcomes for a blood type when randomly selecting an individual from the population. Understanding these proportions is key to solving probability problems involving blood phenotypes.

Working with probabilities often involves understanding the concept of independent events. Two events are independent if the outcome of one does not affect the outcome of the other. In the context of our exercise, this means that knowing the blood phenotype of one individual gives us no information about the blood phenotype of another.When dealing with probabilities involving independent events, we use the multiplication rule. If the events are independent, the probability that both occur is the product of their individual probabilities. For example, if we want to know the chance of both individuals having the O blood phenotype, we calculate:\[ P(O_1 \, \text{and} \, O_2) = P(O_1) \times P(O_2) = 0.44 \times 0.44 = 0.1936 \]This multiplication rule is fundamental when dealing with scenarios that require evaluating the likelihood of multiple independent events happening together.

Matching probabilities involve calculating the likelihood that two separate events result in the same outcome. In the context of blood phenotypes, we are interested in the probability that two randomly selected individuals share the same blood type. To find this probability, we sum the probabilities of each possible matching scenario:

• Both having phenotype A: \(0.42^2 = 0.1764\)
• Both having phenotype B: \(0.10^2 = 0.01\)
• Both having phenotype AB: \(0.04^2 = 0.0016\)
• Both having phenotype O: \(0.44^2 = 0.1936\)

Adding these probabilities gives the total likelihood of a match: \[ P(\text{match}) = 0.1764 + 0.01 + 0.0016 + 0.1936 = 0.3816 \]This approach shows how combining separate probabilities provides insights into the overall chance of certain events co-occurring within a defined context, like blood types in this instance.