Question

“and” and “or” events

1. let the sample space, $s = {a,b,c,d,e,f,g,1,2,3,4,5,6}$, 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?
c. find the event a or b.
d. what is the probability of obtaining event a or b?
e. what is the complement of event a?
f. find $p(a|b)$

Explanation:

Step1: Recall intersection formula

The intersection of two events \(A\) and \(B\), \(A\cap B\), is the set of elements that are in both \(A\) and \(B\). Given \(A = \{b,c,d,g,4,6\}\) and \(B=\{a,b,c,d,f,1,2,6\}\), we find common elements.

Step2: Determine \(A\cap B\)

\(A\cap B=\{b,c,d,6\}\)

Step3: Recall probability formula for intersection

The probability \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}\), where \(n(A\cap B)\) is the number of elements in \(A\cap B\) and \(n(S)\) is the number of elements in the sample - space \(S\). Here, \(n(A\cap B) = 4\) and \(n(S)=13\), so \(P(A\cap B)=\frac{4}{13}\approx0.3077\)

Step4: Recall union formula

The union of two events \(A\) and \(B\), \(A\cup B\), is the set of elements that are in \(A\) or \(B\) (or both). \(A\cup B=\{a,b,c,d,f,g,1,2,4,6\}\)

Step5: Recall probability formula for union

\(P(A\cup B)=\frac{n(A\cup B)}{n(S)}\), with \(n(A\cup B) = 10\) and \(n(S)=13\), so \(P(A\cup B)=\frac{10}{13}\approx0.7692\)

Step6: Recall complement formula

The complement of event \(A\), \(\overline{A}\), is the set of elements in the sample - space \(S\) that are not in \(A\). So \(\overline{A}=\{a,e,f,1,2,3,5\}\)

Step7: Recall conditional - probability formula

The conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). First, \(P(B)=\frac{n(B)}{n(S)}=\frac{8}{13}\), and \(P(A\cap B)=\frac{4}{13}\). Then \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{4}{13}}{\frac{8}{13}}=\frac{4}{8} = 0.5\)

Answer:

a. \(A\cap B=\{b,c,d,6\}\)
b. \(P(A\cap B)=\frac{4}{13}\approx0.3077\)
c. \(A\cup B=\{a,b,c,d,f,g,1,2,4,6\}\)
d. \(P(A\cup B)=\frac{10}{13}\approx0.7692\)
e. \(\overline{A}=\{a,e,f,1,2,3,5\}\)
f. \(P(A|B)=0.5\)