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Question: The probability that a person will hit a target in shooting practice is 0.3. If he shoots 10 times, ...

The probability that a person will hit a target in shooting practice is 0.3. If he shoots 10 times, then the probability of his shooting the target is
A.1
B.1(0.7)101 - {\left( {0.7} \right)^{10}}
C.(0.7)10{\left( {0.7} \right)^{10}}
D.(0.3)10{\left( {0.3} \right)^{10}}

Explanation

Solution

If p and q be the probability of success and failure of an event respectively. Then the probability of that event getting ‘x’ success in an n-Bernoulli’s trial where (ini \leqslant n), then P(Ax)=nCxpxqnxP\left( {{A_x}} \right) = {}^n{C_x}{p^x}{q^{n - x}}.

Complete step-by-step answer:
We do this problem using binomial law, which is, if Ax denotes the event of getting exactly x success in an n-Bernoulli’s trial where (ini \leqslant n), then P(Ax)=nCxpxqnxP\left( {{A_x}} \right) = {}^n{C_x}{p^x}{q^{n - x}}, where p and q denotes the probability of success and failure of that event respectively and p+q=1p + q = 1.
Here, p = the probability that a person will hit the target in shooting practice = 0.3
And, q = the probability that a person will not hit a target in shooting practice=1p=10.3=0.7 = 1 - p = 1 - 0.3 = 0.7
Here, the total number of shoots is 10 i.e. n=10.
\therefore The probability that out of 10 shoots none will hit a target=10C0p0q100=q10=(0.7)10 = {}^{10}{C_0}{p^0}{q^{10 - 0}} = {q^{10}} = {\left( {0.7} \right)^{10}}.
Therefore, the probability of shooting target is 1(0.7)101 - {\left( {0.7} \right)^{10}}
Hence, option (B) is correct.

Note: Another Method –
Let, A be the event that a person will hit a target in shooting practice.
Then the probability of A is given, which is P(A)=0.3P(A) = 0.3.
Now, the probability that a person will not hit a target in shooting practice is P(Aˉ)=1P(A)=10.3=0.7P\left( {\bar A} \right) = 1 - P\left( A \right) = 1 - 0.3 = 0.7
[ We know P\left( A \right) + P\left( {\bar A} \right) = P\left( X \right) = 1$$$$i.e.,P\left( {\bar A} \right) = 1 - P\left( A \right) ]
Now, the probability that out of 10 shoots none will hit a target is P(Aˉ),P(Aˉ),.....uptoP\left( {\bar A} \right),P\left( {\bar A} \right),.....upto10 times = (0.7)10{\left( {0.7} \right)^{10}}
Therefore, the probability of shooting the target =1(0.7)101 - {\left( {0.7} \right)^{10}}