Question

A can hit a target 4 times in 5 shots, B can hit it 3 times in 4 shots and C can hit it 2 times in 3 shots. Calculate the probability that,
1, A, B, C all can hit the target.
2. B, C can hit and A cannot hit.
3. Any two of A, B and C will hit the target
4. None of them will hit the target.

Answer 1

Step-by-step explanation:

A can hit a target 4 times in 5 shots, Probability that A can hit the target

$P(A)=\frac{4}{5}$

Probability that A miss the target

$P(A')=\frac{1}{5}$

B can hit it 3 times in 4 shots

Probability that B can hit the target

$P(B)=\frac{3}{4}$

Probability that B can miss the target

$P(B')=\frac{1}{4}$

and C can hit it 2 times in 3 shots.

Probability that C can hit the target

$P(C)= \frac{2}{3}$

Probability that C can miss the target

$P(C')= \frac{1}{3}$

1)A, B, C all can hit the target.

Ans.

$P(ABC)=P(A)P(B)P(C)\\\\=\frac{4}{5}\times\frac{3}{4} \times\frac{2}{3} \\\\ P(ABC)=\frac{2}{5}$

2. B, C can hit and A cannot hit.

Ans.

$P(A'BC)=P(A')P(B)P(C)\\\\=\frac{1}{5}\times\frac{3}{4} \times\frac{2}{3} \\\\ P(A'BC)=\frac{1}{10}$

3. Any two of A, B and C will hit the target

Ans.

$P(A'BC)+P(AB'C)+P(ABC')=\\\\P(A')P(B)P(C)+P(A)P(B')P(C)+P(A)P(B)P(C')\\\\=\frac{1}{5}\times\frac{3}{4} \times\frac{2}{3}+\frac{4}{5}\times\frac{1}{4} \times\frac{2}{3}+ \frac{4}{5}\times\frac{3}{4} \times\frac{1}{3}\\\\=\frac{6}{60}+\frac{8}{60}+\frac{12}{60} =\frac{26}{60}\\\\=\frac{13}{30}$

4. None of them will hit the target.

Ans.

$P(A'B'C')=P(A')P(B')P(C')\\\\=\frac{1}{5}\times\frac{1}{4} \times\frac{1}{3} \\\\ P(A'B'C')=\frac{1}{60}$

Hope it helps you.