Question

Question 1

Given the following information about events A, B, and C, determine which pairs of events, if any, are independent and which pairs are mutually exclusive.

P(A)=0.26 , P(B)=0.5 , P(C)=0.45 . P(A|B)=0.26 , P(B|C)= 0 , P(C|A)=0.26

Select all correct answers.

Select all that apply:

B and C are independent
A and C are mutually exclusive
A and B are independent
A and C are independent
B and C are mutually exclusive
A and B are mutually exclusive

Question 2

Let E be the event that a randomly chosen person exercises. Let D be the event that a randomly chosen person is on a diet. Identify the answer which expresses the following with correct notation: Of all the people who exercise, the probability that a randomly chosen person is on a diet.

Select the correct answer below:

P(D) AND P(E)
P(E AND D)
P(E|D)
P(D|E)

Answer 1

The only correct answer is:

A and B are independent

A and C are independent

The other options are incorrect.

Given the following information about events A, B, and C:

P(A) = 0.26

P(B) = 0.5

P(C) = 0.45

P(A|B) = 0.26

P(B|C) = 0

P(C|A) = 0.26

We can determine which pairs of events, if any, are independent and which pairs are mutually exclusive.

Independence

Two events are independent if the probability of one event occurring does not affect the probability of the other event occurring. In mathematical terms, events A and B are independent if and only if:

P(A|B) = P(A)

Similarly, events A and C are independent if and only if:

P(A|C) = P(A)

and events B and C are independent if and only if:

P(B|C) = P(B)

Using the given probabilities, we can calculate:

P(A|B) = 0.26

P(A) = 0.26

P(A|C) = 0.26

P(A) = 0.26

P(B|C) = 0

P(B) = 0.5

Since P(A|B) = P(A), P(A|C) = P(A), and P(B|C) ≠ P(B), we can conclude that:

A and B are independent

A and C are independent

B and C are not independent

Mutual Exclusivity

Two events are mutually exclusive if they cannot both occur at the same time. In mathematical terms, events A and B are mutually exclusive if and only if:

P(A ∩ B) = 0

where P(A ∩ B) is the probability of the intersection of events A and B.

Similarly, events A and C are mutually exclusive if and only if:

P(A ∩ C) = 0

and events B and C are mutually exclusive if and only if:

P(B ∩ C) = 0

We can calculate the probabilities of the intersections of the events using the given probabilities:

P(A ∩ B) = P(A) * P(B|A) = 0.26 * 0.26 = 0.0676

P(A ∩ C) = P(A) * P(C|A) = 0.26 * 0.26 = 0.0676

P(B ∩ C) = P(B) * P(C|B) = 0.5 * 0 = 0

Since P(A ∩ B) > 0, P(A ∩ C) > 0, and P(B ∩ C) = 0, we can conclude that:

A and B are not mutually exclusive

A and C are not mutually exclusive

B and C are mutually exclusive

Therefore, the only correct answer is:

A and B are independent

A and C are independent

The other options are incorrect.

Answer 2

Answer with Step-by-step explanation:

1.We are given that three events A, B and C.

P(A)=0.26

P(B)=0.5

P(C)=0.45

P(A/B)=0.26

P(B/C)=0

P(C/A)=0.26

When two events A and B are independent then

`P(A\cap B)=P(A)\cdot P(B)`

If two events are mutually exclusive then

`P(A\cap B)=0`

We know that
`P(A/B)=(P(A\cap B))/(P(B))`

`P(A\cap B)=P(A/B)* P(B)`

`p(A\cap B)=0.26* 0.5=0.13`

`P(A)* P(B)=0.26* 0.5=0.13`

Hence,
`P(A\cap B)=P(A)\cdot P(B)`

Therefore, event A and B are independent.

`P(B\cap C)=0* 0.45=0`

Therefore, events B and C are mutually exclusive.

`P(A\cap C)=0.26* 0.26=0.0676`

`P(A)* P(C)=0.26* 0.45=0.117`

`P(A\cap C)\\eq P(A)\cdot P(C)`

Hence, event A and C are neither independent nor mutually exclusive.

Answer: A and B are independent

B and C are mutually exclusive.

2.Let E be the event that randomly chosen person exercises and D be the event that a randomly chosen person is on a diet.

According to question

We have to find P(D/E).

Answer : P(D/E)