Question

event a: rolling an odd number
event b: rolling a number less than 5
event a and b: rolling an odd number and rolling a number less than 5
event a or b: rolling an odd number or rolling a number less than 5
(b) compute the following.
$p(a)+p(b)-p(a \text{ and } b)=\square$
(c) select the answer that makes the equation true.
$p(a)+p(b)-p(a \text{ and } b)=$ select

Explanation:

Step1: Find the probability of event A

When rolling a fair - six - sided die, the sample space $S=\{1,2,3,4,5,6\}$, $n(S) = 6$. For event $A$ (rolling an odd number), $A=\{1,3,5\}$, $n(A)=3$. So, $P(A)=\frac{n(A)}{n(S)}=\frac{3}{6}=\frac{1}{2}$.

Step2: Find the probability of event B

For event $B$ (rolling a number less than 5), $B = \{1,2,3,4\}$, $n(B)=4$. So, $P(B)=\frac{n(B)}{n(S)}=\frac{4}{6}=\frac{2}{3}$.

Step3: Find the probability of $A$ and $B$

For event $A$ and $B$ (rolling an odd number and a number less than 5), $A\cap B=\{1,3\}$, $n(A\cap B)=2$. So, $P(A\cap B)=\frac{n(A\cap B)}{n(S)}=\frac{2}{6}=\frac{1}{3}$.

Step4: Calculate $P(A)+P(B)-P(A\cap B)$

Substitute the values of $P(A)$, $P(B)$ and $P(A\cap B)$ into the formula:
\[

$$\begin{align*} P(A)+P(B)-P(A\cap B)&=\frac{1}{2}+\frac{2}{3}-\frac{1}{3}\\ &=\frac{3 + 4-2}{6}\\ &=\frac{5}{6} \end{align*}$$

\]

Answer:

$\frac{5}{6}$