Question

a random experiment consists of rolling two fair 5 - sided dice - a red die and a blue die (each die has 5 sides numbered from 1 to 5). consider the following events.
a: the sum is an odd number
b: at least one of the (dies sides) is a 1.
find the following probabilities.
$p(a)=$ (if rounding, use at least four digits after the decimal in your answer)
$p(a^{c})=$ (if rounding, use at least four digits after the decimal in your answer)
$p(acap b)=$ (if rounding, use at least four digits after the decimal in your answer)
$p(a^{c}cap b)=$ (if rounding, use at least four digits after the decimal in your answer)
$p(acup b)=$ (if rounding, use at least four digits after the decimal in your answer)

Explanation:

Step1: Find total number of outcomes

The total number of outcomes when rolling two 5 - sided dice is $n(S)=5\times5 = 25$.

Step2: Define event A

Event A is that the sum of the two dice is an odd number. The sum of two numbers is odd if one is even and the other is odd.
The number of ways to get one even and one odd:
If the red die is even (2 or 4) and the blue die is odd (1, 3, 5), there are $2\times3 = 6$ ways.
If the red die is odd (1, 3, 5) and the blue die is even (2, 4), there are $3\times2=6$ ways. So $n(A)=6 + 6=12$. Then $P(A)=\frac{n(A)}{n(S)}=\frac{12}{25}=0.4800$.

Step3: Find $P(A^{c})$

Since $A^{c}$ is the complement of A, $P(A^{c})=1 - P(A)=1-0.4800 = 0.5200$.

Step4: Define event B

Event B is that at least one of the dice shows a 1. The number of outcomes in B: $(1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(3,1),(4,1),(5,1)$, so $n(B)=9$.

Step5: Find $P(A\cap B)$

The outcomes in $A\cap B$ are $(1,2),(1,4),(2,1),(4,1)$. So $n(A\cap B)=4$, and $P(A\cap B)=\frac{n(A\cap B)}{n(S)}=\frac{4}{25}=0.1600$.

Step6: Find $P(A^{c}\cap B)$

The outcomes in $A^{c}\cap B$ are $(1,1),(1,3),(1,5),(3,1),(5,1)$. So $n(A^{c}\cap B)=5$, and $P(A^{c}\cap B)=\frac{n(A^{c}\cap B)}{n(S)}=\frac{5}{25}=0.2000$.

Step7: Find $P(A\cup B)$

Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. We know $P(A) = 0.4800$, $P(B)=\frac{9}{25}=0.3600$ and $P(A\cap B)=0.1600$. Then $P(A\cup B)=0.4800 + 0.3600-0.1600=0.6800$.

Answer:

$P(A)=0.4800$
$P(A^{c})=0.5200$
$P(A\cap B)=0.1600$
$P(A^{c}\cap B)=0.2000$
$P(A\cup B)=0.6800$