Question

you roll two six - sided fair dice.
a. let a be the event that the first die is even and the second is a 2, 3, 4 or 5.
p(a) = $square$ round your answer to four decimal places.
b. let b be the event that the sum of the two dice is a 7.
p(b) = $square$ round your answer to four decimal places.
c. are a and b mutually exclusive events?
$\bigcirc$ no, they are not mutually exclusive
$\bigcirc$ yes, they are mutually exclusive
d. are a and b independent events?
$\bigcirc$ they are not independent events
$\bigcirc$ they are independent events
hint: mutually exclusive $\boxed{}$
video on probability $\boxed{}$ +

Explanation:

Step1: Total possible outcomes

Total outcomes when rolling two dice: \(6 \times 6 = 36\).

Step2: Calculate P(A)

Event A: First die even (2,4,6: 3 options), second die 2-5 (4 options). Favorable outcomes: \(3 \times 4 = 12\). \(P(A) = \frac{12}{36} = \frac{1}{3} \approx 0.3333\).

Step3: Calculate P(B)

Event B: Sum=7. Favorable pairs: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) (6 outcomes). \(P(B) = \frac{6}{36} = \frac{1}{6} \approx 0.1667\).

Step4: Check mutual exclusivity (A and B)

Common outcomes: (2,5) (first even, second 5; sum=7) and (4,3) (first even, second 3; sum=7). Thus, A and B share outcomes.

Step5: Check independence (A and B)

\(P(A \cap B)\): 2 common outcomes. \(P(A \cap B) = \frac{2}{36} = \frac{1}{18}\). \(P(A) \times P(B) = \frac{1}{3} \times \frac{1}{6} = \frac{1}{18}\). Since \(P(A \cap B) = P(A)P(B)\), events are independent.

Answer:

a. 0.3333
b. 0.1667
c. No, they are not Mutually Exclusive
d. They are Independent events