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An unbiased coin is tossed. If the result is head, a pair of unbiased dice is rolled and the number obtained by adding the number on the two faces is noted. If the result is a tail, a card from a well-shuffled pack of 11 cards numbered 2, 3, 4.... .., 12 is picked & the number on the card is noted. What is the probability that the number noted is 7 or 8?

An unbiased coin is tossed. If the result is head, a pair of unbiased dice is rolled and the number obtained by adding the number on the two faces is noted. If the result is a tail, a card from a well-shuffled pack of 11 cards numbered 2, 3, 4.... .., 12 is picked & the number on the card is noted. What is the probability that the number noted is 7 or 8? Correct Answer 193/792

Calculation:

⇒ Let us define the events

⇒ A : head appears.

⇒ B : Tail appears

⇒ C : 7 or 8 is noted.

⇒ We have to find the probability of C i.e. P (C)

⇒ P(C) = P(A) P (C/A) + P(B) P(C/B)

⇒ Now we calculate each of the constituents one by one

⇒ P(A) = probability of appearing head = 1/2

⇒ P(C/A) = Probability that event C takes place i.e. 7 or 8 being noted when the head has already appeared. (If something has already happened then it becomes certain, i.e. now it is certain that head has appeared we have to certainly roll a pair of unbiased dice).

⇒ 11/36 (since (6, 1) (1, 6) (5, 2) (2, 5) (3, 4) (4, 3) (6, 2) (2, 6) (3, 5) (5, 3) (4, 4) i.e. 11 favourable cases and of course 6 × 6 = 36 total number of cases)

⇒ Similarly, P(B) = 1/2

⇒ P(B/C) = 2/11 (Two favorable cases (7 and 8) and 11 total number of cases).

⇒ Hence, P(C) = 1/2 × 11/36 + 1/2 × 2/11 = 193/792


Additional Information

⇒ The probability of A∩B, (i.e. of B) in the sample space A is m12 / m1.  This is the probability of B under the assumption that A takes place. It is denoted by P(B/A) and is called the conditional probability of B given that A takes place.

⇒ Therefore, P(B/A) = m12 / m1  = n(A ∩ B)/n(A), provided n(A) ≠ 0.

⇒ Similarly, P(A/B) = m12 / m2  = n(A ∩ B)/n(B), provided n(B) ≠ 0.

⇒ Two events A and B are said to be independent if P(A/B) = P(A) and P(B/A) = P(B).

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