Question

you roll two six - sided fair dice.
a. let a be the event that either a 3 or 4 is rolled first followed by an odd number. p(a) = round your answer to four decimal places.
b. let b be the event that the sum of the two dice is at most 7. p(b) = round your answer to four decimal places.
c. are a and b mutually exclusive events? no, they are not mutually exclusive yes, they are mutually exclusive
d. are a and b independent events? they are independent events they are not independent events
hint: mutually exclusive
video on probability +

Explanation:

Step1: Calculate total number of outcomes

When rolling two six - sided dice, the total number of outcomes is $n(S)=6\times6 = 36$.

Step2: Calculate $P(A)$

The first die shows 3 or 4 (2 possibilities), and the second die shows an odd number (3 possibilities: 1, 3, 5). So the number of elements in event $A$ is $n(A)=2\times3 = 6$. Then $P(A)=\frac{n(A)}{n(S)}=\frac{6}{36}\approx0.1667$.

Step3: Calculate $P(B)$

The pairs of dice rolls where the sum is at most 7 are: $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(5,1),(5,2),(6,1)$. The number of elements in event $B$ is $n(B) = 21$. Then $P(B)=\frac{n(B)}{n(S)}=\frac{21}{36}\approx0.5833$.

Step4: Check mutual - exclusivity

Two events are mutually exclusive if $A\cap B=\varnothing$. We can find pairs that are in both $A$ and $B$, for example, $(3,1),(3,3),(4,1),(4,3)$. So $A$ and $B$ are not mutually exclusive.

Step5: Check independence

Two events $A$ and $B$ are independent if $P(A\cap B)=P(A)\times P(B)$. First, find $A\cap B$. The elements of $A\cap B$ are $(3,1),(3,3),(4,1),(4,3)$, so $n(A\cap B) = 4$ and $P(A\cap B)=\frac{4}{36}\approx0.1111$. Also, $P(A)\times P(B)=\frac{6}{36}\times\frac{21}{36}=\frac{126}{1296}\approx0.0972$. Since $P(A\cap B)
eq P(A)\times P(B)$, $A$ and $B$ are not independent.

Answer:

a. $0.1667$
b. $0.5833$
c. No, they are not Mutually Exclusive
d. They are not Independent events