Questions: The range and parity (even or odd) for the history of a die's rolls are related. Find the probability of selecting a roll that was in the upper half and was even.

The range and parity (even or odd) for the history of a die's rolls are related. Find the probability of selecting a roll that was in the upper half and was even.

Transcript text: The range and parity (even or odd) for the history of a die's rolls are related. Find the probability of selecting a roll that was in the upper half and was even.

Solution

Solution Steps

Step 1: Identify the Given Probabilities

Probability of rolling in the upper half: \( P(\text{Upper Half}) = 0.61 \)
Probability of rolling in the lower half: \( P(\text{Lower Half}) = 0.39 \)
Probability of rolling an even number given it is in the upper half: \( P(\text{Even} | \text{Upper Half}) = 0.72 \)

Step 2: Apply the Multiplication Rule for Joint Probability

To find the probability of rolling a number that is both in the upper half and even, we use the multiplication rule for joint probability: \[ P(\text{Upper Half} \cap \text{Even}) = P(\text{Upper Half}) \times P(\text{Even} | \text{Upper Half}) \]

Step 3: Calculate the Joint Probability

Substitute the given probabilities into the formula: \[ P(\text{Upper Half} \cap \text{Even}) = 0.61 \times 0.72 \]

Step 4: Perform the Multiplication

\[ P(\text{Upper Half} \cap \text{Even}) = 0.4392 \]