Question

a standard six - sided die is rolled. which of the following calculations is used to determine if the events roll a number greater than 3 and roll an even number are independent?

Explanation:

Step1: Determine the probability of rolling a number greater than 3

The numbers greater than 3 on a six - sided die are 4, 5, 6. So there are 3 favorable outcomes out of 6 total outcomes. The probability \(P(A)=\frac{3}{6}=\frac{1}{2}\).

Step2: Determine the probability of rolling an even number

The even numbers on a six - sided die are 2, 4, 6. So there are 3 favorable outcomes out of 6 total outcomes. The probability \(P(B)=\frac{3}{6}=\frac{1}{2}\).

Step3: Determine the probability of rolling a number greater than 3 and even

The numbers that are greater than 3 and even are 4, 6. So there are 2 favorable outcomes out of 6 total outcomes. The probability \(P(A\cap B)=\frac{2}{6}=\frac{1}{3}\).

Step4: Check the independence formula \(P(A\cap B)=P(A)\times P(B)\)

If \(P(A)\times P(B)=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}
eq\frac{1}{3}\), the events are not independent. But if we use the formula \(P(A\cap B) = P(A)\times P(B)\) correctly, for two independent events \(A\) and \(B\), we need to calculate \(P(A)\), \(P(B)\) and \(P(A\cap B)\) and check the equality.

Answer:

The formula to check independence is \(P(A\cap B)=P(A)\times P(B)\). Let \(A\) be the event of rolling a number greater than 3 (\(P(A)=\frac{3}{6}=\frac{1}{2}\)), \(B\) be the event of rolling an even number (\(P(B)=\frac{3}{6}=\frac{1}{2}\)), and \(A\cap B\) be the event of rolling a number greater than 3 and even (\(P(A\cap B)=\frac{2}{6}=\frac{1}{3}\)). Since \(\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}
eq\frac{1}{3}\), we use the formula \(P(A\cap B) = P(A)\times P(B)\) to check independence.