Question

1. let the sample - space, s = {a,b,c,d,e,f,1,2,3,4,5,6}, event a = {b,c,d,g,4,6}, and event b = {a,b,c,d,f,1,2,6}.

a. find the event a and b.
b. what is the probability of obtaining event a and b? p(a and b)=\frac{4}{13}=0.3076
c. find the event a or b. p(a or b)={a,b,c,d,f,g,1,2,4,6}
d. what is the probability of obtaining event a or b?
e. what is the complement of event a?
f. find p(a|b)

1. true or false? if two events are independent, then p(a|b)=p(b).
2. true or false? if events a and b are dependent, then p(a and b)=p(a)*p(b).

Explanation:

Step1: Calculate \(P(A\cap B)\)

The number of elements in \(A\cap B=\{b, c, 6\}\), and the sample - space \(S=\{a, b, c, d, f, 1, 2, 3, 4, 5, 6\}\) with \(n(S) = 11\). So \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}=\frac{3}{11}\approx0.2727\)

Step2: Calculate \(P(A\cup B)\)

\(A = \{b, c, d, g, 4, 6\}\), \(B=\{a, b, c, d, f, 1, 2, 6\}\), \(A\cup B=\{a, b, c, d, f, g, 1, 2, 4, 6\}\), \(n(A\cup B)=10\). So \(P(A\cup B)=\frac{n(A\cup B)}{n(S)}=\frac{10}{11}\approx0.9091\)

Step3: Find the complement of \(A\)

\(\overline{A}=S - A=\{a, f, 1, 2, 3, 5\}\), \(n(\overline{A}) = 6\). So \(P(\overline{A})=\frac{n(\overline{A})}{n(S)}=\frac{6}{11}\approx0.5455\)

Step4: Calculate \(P(A|B)\)

By the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). First, \(P(B)=\frac{n(B)}{n(S)}=\frac{7}{11}\), and \(P(A\cap B)=\frac{3}{11}\). So \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{3}{11}}{\frac{7}{11}}=\frac{3}{7}\approx0.4286\)

Step5: Judge the truth - value of statement 2

If two events are independent, \(P(A|B)=P(A)\). Here, when events are independent, \(P(A|B)
eq P(B)\), so the statement "If two events are independent, then \(P(A|B) = P(B)\)" is False.

Step6: Judge the truth - value of statement 3

If two events \(A\) and \(B\) are dependent, \(P(A\cap B)
eq P(A)\times P(B)\). The formula \(P(A\cap B)=P(A)\times P(B)\) holds for independent events. So the statement "If events \(A\) and \(B\) are dependent, then \(P(A\cap B)=P(A)\times P(B)\)" is False.

Answer:

a. \(P(A\cap B)=\frac{3}{11}\)
b. \(P(A\cup B)=\frac{10}{11}\)
c. \(P(\overline{A})=\frac{6}{11}\)
d. \(P(A|B)=\frac{3}{7}\)
e. False
f. False