Shooters
In a regiment there are piati shooters. The first shooter hits the target with a probability of 42%, and the next with 40%, 21%, 58%, 67%%. Calculate the probability that the target is hit when all shooters fire simultaneously.
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Chris Roberts
Let's denote:
H: Hit
M: Miss
Now, we can represent the probabilities as follows:
Shooter 1: P(H1) = 0.49, P(M1) = 1 - P(H1) = 0.51
Shooter 2: P(H2) = 0.75, P(M2) = 1 - P(H2) = 0.25
Shooter 3: P(H3) = 0.41, P(M3) = 1 - P(H3) = 0.59
Shooter 4: P(H4) = 0.20, P(M4) = 1 - P(H4) = 0.80
Shooter 5: P(H5) = 0.34, P(M5) = 1 - P(H5) = 0.66
Shooter 6: P(H6) = 0.63, P(M6) = 1 - P(H6) = 0.37
Now, we want to find the probability of all shooters hitting the target when shooting at once. We can use the multiplication rule for independent events, which states that the probability of all independent events occurring together is the product of their individual probabilities.
Let's calculate the probability of hitting the target when shooting all six shooters at once:
P(all shooters hit) = P(H1) * P(H2) * P(H3) * P(H4) * P(H5) * P(H6)
P(all shooters hit) = 0.49 * 0.75 * 0.41 * 0.20 * 0.34 * 0.63
P(all shooters hit) ≈ 0.008728 (rounded to six decimal places)
So, the probability of hitting the target when shooting all six shooters at once is approximately 0.008728 or about 0.873%.
H: Hit
M: Miss
Now, we can represent the probabilities as follows:
Shooter 1: P(H1) = 0.49, P(M1) = 1 - P(H1) = 0.51
Shooter 2: P(H2) = 0.75, P(M2) = 1 - P(H2) = 0.25
Shooter 3: P(H3) = 0.41, P(M3) = 1 - P(H3) = 0.59
Shooter 4: P(H4) = 0.20, P(M4) = 1 - P(H4) = 0.80
Shooter 5: P(H5) = 0.34, P(M5) = 1 - P(H5) = 0.66
Shooter 6: P(H6) = 0.63, P(M6) = 1 - P(H6) = 0.37
Now, we want to find the probability of all shooters hitting the target when shooting at once. We can use the multiplication rule for independent events, which states that the probability of all independent events occurring together is the product of their individual probabilities.
Let's calculate the probability of hitting the target when shooting all six shooters at once:
P(all shooters hit) = P(H1) * P(H2) * P(H3) * P(H4) * P(H5) * P(H6)
P(all shooters hit) = 0.49 * 0.75 * 0.41 * 0.20 * 0.34 * 0.63
P(all shooters hit) ≈ 0.008728 (rounded to six decimal places)
So, the probability of hitting the target when shooting all six shooters at once is approximately 0.008728 or about 0.873%.
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