The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed.
The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that
Question: 1
Target is hit
Show Hint
For "at least" problems, always consider calculating the "none" scenario and subtracting it from 1.
Ensure that probabilities of exhaustive events sum to 1 before proceeding.
• Total probability of an event and its complement: \( P(E) + P(E') = 1 \).
• Bernoulli trials for independent events: The outcome of one shot does not affect the other.
• Complementary events: \( P(\text{at least one}) = 1 - P(\text{none}) \).
Step 1: Calculate individual probabilities for hitting and missing
Let \( p \) be the probability of hitting the target and \( q \) be the probability of missing it.
Given that \( p = 3q \).
We know that \( p + q = 1 \).
Substituting \( p \):
\[ 3q + q = 1 \implies 4q = 1 \implies q = 1/4 \]
Then, \( p = 3(1/4) = 3/4 \). Step 2: Calculate the probability that the target is hit
The target is considered "hit" if at least one of the two shots hits the target.
It is easier to calculate the complement: both shots miss.
\( P(\text{both miss}) = q \times q = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \).
\( P(\text{target hit}) = 1 - P(\text{both miss}) = 1 - \frac{1}{16} = \frac{15}{16} \).
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Question: 2
Atleast one shot misses the target.
Show Hint
For "at least" problems, always consider calculating the complementary event.
Ensure that probabilities of exhaustive events sum to 1 before proceeding.
Concept: • Total probability of an event and its complement: \( P(E) + P(E') = 1 \).
• Bernoulli trials for independent events: The outcome of one shot does not affect the other.
• Complementary events: \( P(\text{at least one}) = 1 - P(\text{none}) \).
Step 1: Calculate individual probabilities for hitting and missing
Let \( p \) be the probability of hitting the target and \( q \) be the probability of missing it.
Given that \( p = 3q \).
We know that \( p + q = 1 \).
Substituting \( p \):
\[ 3q + q = 1 \implies 4q = 1 \implies q = 1/4 \]
Then, \( p = 3(1/4) = 3/4 \).
Step 2: Calculate the probability that at least one shot misses the target
At least one miss means the outcome is NOT "both hits".
\( P(\text{both hit}) = p \times p = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \).
\( P(\text{at least one miss}) = 1 - P(\text{both hit}) = 1 - \frac{9}{16} = \frac{7}{16} \).
