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Answer :
The expected number of times the target will be hit is [tex]\( \frac{21}{8} \)[/tex] or 2.625 times. Therefore, on average, we can expect the target to be hit approximately 2.625 times in this scenario.
To find the expected number of times the target will be hit, we need to calculate the expected value for each shooter and then sum them up.
For shooter A:
- Shooter A shoots 3 times.
- The probability of hitting the target on any given shot is [tex]\( \frac{1}{8} \)[/tex].
- We can use the binomial distribution formula to calculate the expected number of hits for shooter A:
[tex]\[ E_A = n_A \times p_A \][/tex]
[tex]\[ E_A = 3 \times \frac{1}{8} = \frac{3}{8} \][/tex]
For shooter B:
- Shooter B shoots 5 times.
- The probability of hitting the target on any given shot is [tex]\( \frac{1}{4} \)[/tex].
- Using the same formula, we find:
[tex]\[ E_B = n_B \times p_B \][/tex]
[tex]\[ E_B = 5 \times \frac{1}{4} = \frac{5}{4} \][/tex]
For shooter C:
- Shooter C shoots 2 times.
- The probability of hitting the target on any given shot is [tex]\( \frac{1}{2} \)[/tex].
- Again, using the same formula, we find:
[tex]\[ E_C = n_C \times p_C \][/tex]
[tex]\[ E_C = 2 \times \frac{1}{2} = 1 \][/tex]
Now, we sum up the expected values for each shooter:
[tex]\[ E_{total} = E_A + E_B + E_C \][/tex]
[tex]\[ E_{total} = \frac{3}{8} + \frac{5}{4} + 1 \][/tex]
[tex]\[ E_{total} = \frac{3}{8} + \frac{10}{8} + \frac{8}{8} \][/tex]
[tex]\[ E_{total} = \frac{21}{8} \][/tex]
So, the expected number of times the target will be hit is [tex]\( \frac{21}{8} \)[/tex]or 2.625 times.
Therefore, on average, we can expect the target to be hit approximately 2.625 times in this scenario.
The complete question is:
Three men a, b, and c shoot at a target. suppose that a shoots three times and the probability that he will hit the target on any given shot is 1/8, b shoots five times and the probability that he will hit the target on any given shot is 1/4, and c shoots twice and the probability that he will hit the target on any given shot is 1/2. what is the expected number of times that the target will be hit?
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