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: Calculate probabilities

• Probability of rolling a number greater than 3 (\(P(A)\)): There are 3 numbers (\(4\), \(5\), \(6\)) out of 6, so \(P(A)=\frac{3}{6}=\frac{1}{2}\).
• Probability of rolling an even number (\(P(B)\)): There are 3 even numbers (\(2\), \(4\), \(6\)) out of 6, so \(P(B)=\frac{3}{6}=\frac{1}{2}\).
• Probability of rolling a number greater than 3 and even (\(P(A\cap B)\)): There are 2 numbers (\(4\), \(6\)) out of 6, so \(P(A\cap B)=\frac{2}{6}=\frac{1}{3}\).

Step2: Check independence

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\).
\(P(A)\times P(B)=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}
eq\frac{1}{3}\)

Answer:

\(\frac{1}{3}
eq\frac{1}{2}\times\frac{1}{2}\) (the option where \(\frac{1}{3}
eq\frac{1}{2}\times\frac{1}{2}\) is the correct one, but since the options are not clearly labeled with letters in the image text, based on the calculation shown above, the key is the comparison \(\frac{1}{3}
eq\frac{1}{2}\times\frac{1}{2}\))