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In probability theory, the distribution of blood types is a fascinating example that demonstrates how certain traits are found in a population. Blood type distribution refers to how common each blood type is in a given group. In Emma and Josh's case, the distribution is as follows: blood type A makes up 40% of the population, B constitutes 10%, AB is 5%, and O is 45%. To find probabilities, these percentages are converted into decimal form: 0.40, 0.10, 0.05, and 0.45 respectively.

Understanding this distribution helps determine how likely it is for any two people to share or differ in blood type. These proportions indicate that blood type O is the most common while AB is the rarest. Such data can be helpful not just in calculating probability but also in medical contexts such as transfusions and disease studies.

Complementary probability involves finding the likelihood of an event by understanding the likelihood of its opposite occurring. In this exercise, we are determining the probability that Emma and Josh have different blood types by looking at the reverse - what if they had the same blood type?

• First, calculate the probability they have the same blood type by squaring each probability and adding them together: \((0.40)^2 + (0.10)^2 + (0.05)^2 + (0.45)^2\).
• This gives you the probability of Emma and Josh having the same blood type.
• This is a classic example of complementary probability, where the probability of them having different blood types is simply one minus the probability of them having the same blood type.

With the principle of complementary probability, calculating the chance of differing blood types becomes straightforward, enhancing our understanding of how probabilities can be derived indirectly.

Calculating probability involves breaking down the event and determining its likelihood numerically. For Emma and Josh, once we understand the probabilities of blood types, we perform the necessary computations.

• First, the probability they have the same blood type is determined by the sum \((0.40)^2 + (0.10)^2 + (0.05)^2 + (0.45)^2 = 0.42\).
• The probability of them having different blood types is calculated as \(1 - 0.42 = 0.58\).

These calculations show that there is a 58% chance that Emma and Josh will have different blood types. This example emphasizes the simplicity and power of probability theory in analyzing real-world scenarios, showing that seemingly complex situations can be understood and quantified with clear numerical expressions.