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}), 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?

Explanation:

Step1: Recall intersection formula

The event "A and B" is the intersection $A\cap B$, which consists of elements common to both A and B. Given $A = \{b,c,d,g,4,6\}$ and $B=\{a,b,c,d,f,1,2,6\}$, we find the common elements.

Step2: Calculate probability of 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 $n(S)=13$.

Step3: Recall union formula

The event "A or B" is the union $A\cup B$, which consists of all elements that are in A or in B or in both.

Step4: Calculate probability of union

First, count the number of elements in $A\cup B$. Then use the formula $P(A\cup B)=\frac{n(A\cup B)}{n(S)}$. For $A\cup B=\{a,b,c,d,f,g,1,2,4,6\}$, $n(A\cup B) = 10$ and $n(S)=13$.

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. $P(A\cup B)=\frac{10}{13}\approx0.7692$