Answer:
1a. 5040
1b. 35
1c. 840
2a. see below
2b. [tex]\frac{1}{4}[/tex]
2c. [tex]\frac{3}{7}[/tex]
2d. 1.39
Step-by-step explanation:
1.
(a) Arranging 7 pictures → order is significant, therefore we use the Permutation Probability.
[tex]\boxed{_nP_r=\frac{n!}{(n-r)!} }[/tex]
n = total number of subjects
r = total number of selected subjects
Given:
n = 7
r = 7
[tex]\displaystyle{_7P_7=\frac{7!}{(7-7)!} }[/tex]
[tex]=7![/tex]
[tex]=5040[/tex]
(b) Picking 4 paintings out of 7 → order is not significant, so we use the Combination Probability.
[tex]\boxed{_nC_r=\frac{n!}{r!(n-r)!} }[/tex]
Given:
n = 7
r = 4
[tex]\displaystyle_7P_4=\frac{7!}{4!(7-4)!}[/tex]
[tex]\displaystyle=\frac{7!}{4!\ 3!} }[/tex]
[tex]\displaystyle=\frac{7\times6\times5}{3!} }[/tex]
[tex]\displaystyle=\frac{7\times6\times5}{3\times2\times1}[/tex]
[tex]=35[/tex]
(c) Arranging 4 out of 7 painting → order is significant → Permutation Probability.
Given:
n = 7
r = 4
[tex]\displaystyle{_7P_4=\frac{7!}{(7-4)!} }[/tex]
[tex]\displaystyle=\frac{7!}{3!}[/tex]
[tex]=7\times6\times5\times4[/tex]
[tex]=840[/tex]
**Notes: using the question (b) data → each 1 of the 35 probabilities has [tex]_4P_4[/tex] arrangements. Then, the total arrangement will be: [tex]35\times _4P_4=35\times 4!=840[/tex]
2.
(a)
∑frequency = 6 + 10 + 7 + 5
= 28
[tex]P(X=0)=\frac{6}{28} =\frac{3}{14}[/tex]
[tex]P(X=1)=\frac{10}{28} =\frac{5}{14}[/tex]
[tex]P(X=2)=\frac{7}{28} =\frac{1}{4}[/tex]
[tex]P(X=3)=\frac{5}{28}[/tex]
Probability Distribution Table
[tex]\begin{array}{c|c|c|c|c}X & 0 & 1 & 2 & 3\\\cline{1-5}P(X) & \frac{3}{14} &\frac{5}{14} &\frac{1}{4} &\frac{5}{28} \end{array}[/tex]
(b) P(X=2) = [tex]\frac{1}{4}[/tex]
(c) P(X>1) = P(X=2) + P(X=3)
[tex]=\frac{1}{4} +\frac{5}{28}[/tex]
[tex]=\frac{3}{7}[/tex]
(d) Estimated value = Expectation (E(X))
[tex]\boxed{E(X)=\Sigma x_i\cdot p_i}[/tex]
[tex]E(X) =0\cdot\frac{3}{14} +1\cdot\frac{5}{14} +2\cdot\frac{1}{4} +3\cdot\frac{5}{28}[/tex]
[tex]=\frac{39}{28}[/tex]
[tex]\approx 1.39[/tex]
