In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards.
Archer A is likely to earn 10 points with a probability of 0·8 and Archer B is likely the earn 10 points with a probability of 0·9.
Based on the above information, answer the following questions:
If both of them hit the Archery target, then find the probability that
(a) exactly one of them earns 10 points.
(b) both of them earn 10 points.

Question

In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards.
Archer A is likely to earn 10 points with a probability of 0·8 and Archer B is likely the earn 10 points with a probability of 0·9.
Based on the above information, answer the following questions:
If both of them hit the Archery target, then find the probability that
(a) exactly one of them earns 10 points.
(b) both of them earn 10 points.

Accepted Answer

Assume, thatP(A)= probability of Archer A is likely to earn 10 points=0.8P(B)= probability of Archer B is likely to earn 10 points=0.9Now, for P’(A)=1-P(A)=1-0.8=0.2 and P’(B)=1-P(B)=1-0.9=0.1(a) Now, the required probability for exactly one of them earns 10 points=P(A)⋅P’(B)+P’(A)⋅P(B)=0.8⋅0.1+0.2⋅0.9=0.08+0.18=0.26(b) Now, the required probability for both of them earn 10 points,∴(∩)=()×()=0.8×0.9=0.72Therefore, (a) Required probability is 0.26,(b) Required probability is 0.72

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