Answer to Question #127769 in Statistics and Probability for Jack

Question

Ali and Baba shoot arrows at a target. The probability that Ali hits the target is 1/4. The probability that Baba
hits the target is 1/3.
.
a) If Ali shot the target 8 times, find the probability that he hit the target for
i) Exactly 2 times.
ii) 3 times or more.
b) Ali and Baba each shot the target 3 times, find the probability that they hit the target with the same
number of arrows each (either 0,1,2 or 3).
c) Ali keeps shooting at the target until he eventually hits the target once. Find the probability that he
takes 5 shots to do so.
d) Baba keeps shooting at the target until he eventually hits the target twice. Find the probability that he
takes 4 shots to do so.

Answer 1

a)

Bernoulli distribution.

Exactly 2 times:

$P(2)=C_8^2\cdot(\frac{1}{4})^2\cdot(1-\frac{1}{4})^{8-2} \approx 0.311$

3 times or more:

$P(0)=C_8^0\cdot(\frac{1}{4})^0\cdot(1-\frac{1}{4})^{8-0} \approx 0.1 \\[0.2cm]$

$P(1)=C_8^1\cdot(\frac{1}{4})^1\cdot(1-\frac{1}{4})^{8-1} \approx 0.267 \\[0.2cm]$

$P(\geq3)=1-P(0)-P(1)-P(2) \approx 1 -0.1 -0.267 -0.311 = 0.322$

b)

Ali shots and Baba shots are independent events. Still Bernoulli distribution.

$P_{Ali}(0)=C_3^0\cdot(\frac{1}{4})^0\cdot(1-\frac{1}{4})^{3-0} \approx 0.422 \\[0.2cm] P_{Ali}(1)=C_3^1\cdot(\frac{1}{4})^1\cdot(1-\frac{1}{4})^{3-1} \approx 0.422 \\[0.2cm] P_{Ali}(2)=C_3^2\cdot(\frac{1}{4})^2\cdot(1-\frac{1}{4})^{3-2} \approx 0.141 \\[0.2cm] P_{Ali}(3)=C_3^3\cdot(\frac{1}{4})^3\cdot(1-\frac{1}{4})^{3-3} \approx 0.016 \\[0.2cm]$

$P_{Baba}(0)=C_3^0\cdot(\frac{1}{3})^0\cdot(1-\frac{1}{3})^{3-0} \approx 0.296 \\[0.2cm] P_{Baba}(1)=C_3^1\cdot(\frac{1}{3})^1\cdot(1-\frac{1}{3})^{3-1} \approx 0.444 \\[0.2cm] P_{Baba}(2)=C_3^2\cdot(\frac{1}{3})^2\cdot(1-\frac{1}{3})^{3-2} \approx 0.222 \\[0.2cm] P_{Baba}(3)=C_3^3\cdot(\frac{1}{3})^3\cdot(1-\frac{1}{3})^{3-3} \approx 0.037 \\[0.2cm]$

$P=P_{Ali}(0)\cdot P_{Baba}(0) + P_{Ali}(1)\cdot P_{Baba}(1) + P_{Ali}(2)\cdot P_{Baba}(2) + P_{Ali}(3)\cdot P_{Baba}(3) \approx \\[0.1cm] \approx 0.422\cdot0.296+0.422\cdot0.444+0.141\cdot0.222+0.016\cdot0.037 \approx \\[0.1cm] \approx 0.344$

c)

Ali takes 5 shots. So he misses 4 times and then hits the traget.

$P=(1-\frac{1}{4})\cdot(1-\frac{1}{4})\cdot(1-\frac{1}{4})\cdot(1-\frac{1}{4})\cdot\frac{1}{4}\approx 0.079$

d)

Baba takes 4 shots. So when he shoots 4th time he hits the target 2nd time. First 3 shots can create any sequence of 1 hit and 2 missings. Bernoulli distribution again.

$P=\left(C_3^1\cdot(\frac{1}{3})^1\cdot(1-\frac{1}{3})^2\right)\cdot\frac{1}{3}\approx0.148$