A red and a blue die are thrown. Both dice are fair. The events A, B, and C are

Question

A red and a blue die are thrown. Both dice are fair. The events A, B, and C are defined as follows:

A: The sum on the two dice is even

B: The sum on the two dice is at least 10

C: The red die comes up 5

(e) Which pairs of events among A, B, and C are independent?

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A biased coin is flipped 10 times. In a single flip of the coin, the probability of heads is 1/3 and the probability of tails is 2/3. The outcomes of the coin flips are mutually independent. What is the probability of each event?

(c) The first roll comes up heads. The rest of the rolls come up tails.

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Explanation / Answer

A: The sum on the two dice is even
B: The sum on the two dice is at least 10
C: The red die comes up 5

A and B: Not independent B restricts the sum, so they are dependent

A and C: Not independent because the sum is restricted by the other dice

B and C: Not independent because if we restrict outcome of one dice, the outcome of ther dice will be restricted when if the sum of two outcomes are 10

To find the probability of the first roll comes up heads and the rest of the rolls come up tails we need to know the number of total number of rolls. let total number of rolls be n, then we have 1 head and n-1 tails
since both events are independent we can say P(The first roll comes up heads. The rest of the rolls come up tails) = P(head) * P(tails)

p(head) = 1/3
p(tails) = p(tail)n-1 = (2/3)n-1

So P(The first roll comes up heads. The rest of the rolls come up tails) = 1/3 * (2/3)n-1

We know n = 10
so P(The first roll comes up heads. The rest of the rolls come up tails) = 1/3 * (2/3)9 = 512/310

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