Question

Although it's hard to be definitive in classifying people as right- or left- handed, some studies suggest that about \(14 \%\) of people are left-handed. Since \(0.14 \times 0.14=0.0196\), the Multiplication Rule might suggest that there's about a \(2 \%\) chance that a brother and a sister are both lefties. What's wrong with that reasoning?

Short answer

The reasoning assumes independence, but handedness may be genetically influenced, making the events dependent.

Step-by-step solution

Understand the Multiplication Rule

The Multiplication Rule states that if two events, A and B, are independent, then the probability of both occurring is the product of their probabilities: \( P(A \text{ and } B) = P(A) \times P(B) \). This rule assumes the two events do not influence each other.

Identify the Events and Probabilities

In this problem, the events are: brother being left-handed (Event A) and sister being left-handed (Event B). Given that \( P(A) = 0.14 \) and \( P(B) = 0.14 \). The original reasoning uses the formula \( P(A \text{ and } B) = 0.14 \times 0.14 = 0.0196 \) or approximately \(2\%\).

Examine the Assumption of Independence

For the Multiplication Rule to apply, the events must be independent, meaning the handedness of the brother has no effect on the handedness of the sister. However, handedness can be influenced by genetic factors, making these events possibly dependent, not independent.

Consider Genetic Factors

Handedness can sometimes run in families due to genetic factors. If one sibling is left-handed, the probability that another sibling is also left-handed can be higher than if they were unrelated, contravening the assumption of independence.

Conclusion

Since the events are not necessarily independent due to potential genetic influences, the simple application of the Multiplication Rule is incorrect. The probability of both siblings being left-handed could be different from \(2\%\).

Key concepts

Multiplication Rule

The Multiplication Rule is a fundamental concept in probability that helps calculate the probability of two separate events happening at the same time. It's expressed as: \[ P(A \text{ and } B) = P(A) \times P(B) \]This equation is only valid if Events A and B are independent. Independence means the outcome of one event doesn't affect the other. For example:

• If you flip a coin twice, the outcome of the first flip doesn't change the outcome of the second flip. This makes them independent events.

But be careful! If the events aren't independent, using the Multiplication Rule directly can give incorrect results. It's important to verify the independence of events before applying this rule to ensure accurate probability calculations.

Independence in Probability

Independence in probability is a key assumption when applying the Multiplication Rule. Two events are independent if the outcome of one event does not change the outcome of the other.

• In practical terms, consider rolling two dice. The result of the first die doesn't impact the result of the second die, making them independent.
• On the other hand, genetic or environmental links can lead events to be dependent. For example, the probability of raining tomorrow depends on today’s weather, making these events dependent.

If events are dependent, their outcomes are connected. For example, in the case of siblings and handedness, genetics could play a role, meaning a brother and sister might have a related probability of being left-handed.

Genetic Influence on Probability

Genetic factors can profoundly influence probabilities, especially in traits that run in families. Certain characteristics, such as handedness, can be influenced by heritable traits. This means:

• If one sibling is left-handed, the probability that another sibling is also left-handed may be different from someone unrelated.
• This kind of probability is called conditional probability, where the outcome of one event affects the probability of another.

Due to these genetic influences, siblings might display similar traits more frequently than randomly related individuals. It's important to account for such factors when using probabilities, ensuring that dependencies are considered in your calculations. This adjustment can prevent overestimations or underestimations of probabilities.