Question

You roll two six-sided fair dice. a. Let A be the event that the first die is odd and the second is a 4, 5, or 6. P(A) =? Round your answer to four decimal places. b. Let B be the event that the sum of the two dice is 10. P(B) =? Round your answer to four decimal places. c. Are A and B mutually exclusive events?d. Are A and B independent events?

Answer 1

Answer:

a. $\dfrac{1}{4}$

b. $\dfrac{1}{12}$

c. Not mutually exclusive.

d. Not independent events

Step-by-step explanation:

a. When two six-sided fair dice are rolled one time,

Total possible outcomes, n(S) = 6 × 6 = 36,

If A be the event that the first die is odd and the second is a 4, 5, or 6

Then, A = {(1, 4), (3, 4), (5, 4), (1, 5), (3, 5), (5, 5), (1, 6), (3, 6), (5, 6)}

i.e. n(A) = 9,

$\because \text{Probability}=\frac{\text{Favourable outcomes}}{\text{Total outcomes}}$

Thus, the probability of event A,

$P(A) = \frac{n(A)}{n(S)}=\frac{9}{36}=\frac{1}{4}$

b. if B be the event that the sum of the two dice is 10,

Then B = {(4, 6), (5, 5), (6, 4)}

i.e. n(B) = 3,

Thus, the probability of event B,

$P(B) = \frac{n(B)}{n(S)}=\frac{3}{36}=\frac{1}{12}$

c. Two event are called mutually exclusive events,

If they are disjoint,

i.e. A and B are mutually exclusive,

If A ∩ B = ∅

∵ {(1, 4), (3, 4), (5, 4), (1, 5), (3, 5), (5, 5), (1, 6), (3, 6), (5, 6)} ∩ {(4, 6), (5, 5), (6, 4)}

= {(5, 5)} ≠ ∅

So, they are not mutually exclusive events.

d. Two events are called independent if the occurrence of one event does not affect the occurrence of other event.

Also, A and B are independent events,

If P(A ∩ B) = P(A) × P(B)

n(A∩ B) = 1,

$\implies P(A\cap B)=\frac{n(A\cap B)}{n(S)}=\frac{1}{36}$

∵ $\frac{1}{4}\times \frac{1}{12}=\frac{1}{48}\neq \frac{1}{36}$

Hence, they are not independent events.