Thank you for visiting Three riflemen A B and C take turns shooting at a target The first rifleman to hit the target wins 2002 dollars A shoots first. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Rifleman A's expected winnings are 1502 dollars.
What is probability?
The probability of an event occurring is defined by probability. There are many instances in real life where we may need to make predictions about how something will turn out.
Let X be the random variable representing the amount of money rifleman A wins. We want to find E[X], the expected value of X.
If rifleman A hits the target on his first turn, he wins 2002 dollars. This happens with probability 0.5.
If rifleman A misses on his first turn and rifleman B misses on his turn, the game returns to rifleman A with no money won or lost. The probability of this happening is (0.5)(0.5) = 0.25.
If rifleman A misses on his first turn, rifleman B hits on his turn, and rifleman C misses on his turn, the game returns to rifleman A with no money won or lost. The probability of this happening is (0.5)(0.5)(0.5) = 0.125.
If rifleman A misses on his first turn, rifleman B misses on his turn, and rifleman C hits on his turn, the game returns to rifleman A with a potential win of 2002 dollars. The probability of this happening is (0.5)(0.5)(0.5) = 0.125.
If rifleman A misses on his first turn, rifleman B hits on his turn, and rifleman C hits on his turn, the game returns to rifleman A with a potential win of 2002 dollars. The probability of this happening is (0.5)(0.5)(0.5) = 0.125.
Therefore, we can write the expected value of X as:
E[X] = 0.5(2002) + 0.25(0) + 0.125(0) + 0.125(2002) + 0.125(2002)
E[X] = 1001 + 250.5 + 250.5
E[X] = 1502
So, rifleman A's expected winnings are 1502 dollars.
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Rewritten by : Jeany
Main Answer: A's expected winnings are 2002 dollars.
Supporting Question and Answer:
What is probability?
Probability is the measure of the likelihood or chance of an event occuring.It is a mathematical concept used to quantify uncertainly and to make predictions about the likelihood of future events.Probability is expressed as a number between 0 and 1[ 0 indicates that an event is impossible and 1 indicates that an event is certain to occur.
Body of the Solution: Let the expected winnings of A are E.
A can win in two ways:
- A hits the target on his first turn, then the probability of A is 0.5 and in this case, he wins 2002 dollars.
- A misses the target on his first turn and his expected winnings are the same as his overall expected winnings,then it is B's turn to shoot, and so on until the cycle repeats , after which A's expected winnings are E again.The probability of A missing the target on his first turn is 0.5.
In this case,the cycle repeats until A eventually hits the target and A's expected winnings are the same as his overall expected winnings.Therefore, E can be expressed as follows:
E=0.5*2002+0.5*E
0.5*E=2002*0.5
E=2002
Final Answer: Therefore, A's expected winnings are 2002 dollars.
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