Question

Determine whether the following individual events are overlapping or non-overlapping. Then find the probability of the combined event.Getting a sum of either 5 or 9 on a roll of two dice Choose the correct answer below and, if necessary, fill in the answer box to complete your choice.(Type an integer or a simplified fraction.)A.The individual events are non-overlapping. The probability of the combined event is   enter your response here.B.The individual events are overlapping. The probability of the combined event is

Answer 1

Given:

Two dice are rolled.

Required:

Probability of getting a sum of 5 or 9 on a roll of two dice.

Are these two events overlapping or non overlapping.

Step-by-step explanation:

The sample space of the event when two dice is rolled is given by

`\begin{gathered} S=\lbrace(1,1),(1,2),(1,3),(1,4),(1,5),(1,6), \\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6), \\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), \\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6), \\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6), \\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\rbrace \end{gathered}`

This is the sample space when two dice are rolled

`n(S)=36`

Event 1 : getting a sum of 5 when two dice are rolled

`\begin{gathered} E_1=\lbrace(1,4),(2,3),(3,2),(4,1)\rbrace \\ \\ n(E_1)=4 \end{gathered}`

Probability of getting a sum of 5 when two dice are rolled is the ratio of number of outcomes in event E₁ to the number of outcomes in sample space.

`P(E_1)=(n(E_1))/(n(S))=(4)/(36)=(1)/(9)`

Now the second event is getting a sum of 9 when two dice are rolled

`\begin{gathered} E_2=\lbrace(3,6),(4,5),(5,4),(6,3)\rbrace \\ \\ n(E_2)=4 \end{gathered}`

Probability of getting a sum of 9 when two dice are rolled is the ratio of number of outcomes in event E₂ to the number of outcomes in sample space.

`P(E_2)=(n(E_2))/(n(S))=(4)/(36)=(1)/(9)`

If we see both the events there are no sub events that are common in E₁ and E₂

So the events are non-overlapping.

The probability of combined event is given as the sum of probability of both the events

`\begin{gathered} P(combined\text{ }event)=P(E_1)+P(E_2) \\ \\ P(combined\text{ }event)=(1)/(9)+(1)/(9)=(2)/(9) \end{gathered}`

Final answer:

The individual events are non-overlapping.

The probability of the combined event is 2/9.