Mathematics : asked on 18080980
 12.06.2021

The events A and B are mutually exclusive. If P(A)= 0.1 and P(B) = 0.5, what is P(A and B)?

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23.03.2022, solved by verified expert
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For mutually exclusive event. There is nothing common between A and B.

Therefore 

P(A and B) = 0 

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Mathematics
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P Answered by PhD
The answer is D. 0.6

P(A or B) = P(A) + P(B) - P(A and B)
When the events are mutually exclusive, that means they cannot occur together:
P(A and B) is empty set, thus = 0
P(A) = 0.25
P(B) = 0.35
P(A or B) = 0.25 + 0.35 - 0 = 0.6
Mathematics
Step-by-step answer
P Answered by PhD
The answer is D. 0.6

P(A or B) = P(A) + P(B) - P(A and B)
When the events are mutually exclusive, that means they cannot occur together:
P(A and B) is empty set, thus = 0
P(A) = 0.25
P(B) = 0.35
P(A or B) = 0.25 + 0.35 - 0 = 0.6
Mathematics
Step-by-step answer
P Answered by PhD

We want to select the statements that are axioms of probability.

We will see that the correct options are C, D, and E.

The easier thing to do, is writing the axioms and see which statements relate to those.

The axioms of probability are:

The probability of an event is a non-negative real number and equal to or smaller than 1. The probability of one of all the possible events to happen is 1, this also means that the sum of all the probabilities of the individual events is equal to 1.The probability (assuming additivity) of one event or other happening is equal to the sum of the probabilities.

Now that we know the axioms, let's see if the statements relate to them or not.

A) The probability of two events occurring is always greater than 0.

false, if we take two events with a probability of zero, then the probability of these two events occurring is zero.

B) The probability of an event A is the number of outcomes in A divided by the number of outcomes in the sample space.

This is false, this is only true when all the outcomes in the sample space have the same probability of happening.

C) The probability of an event is a number that is at least 0 and no more than 1.

True, by the first axiom.

D) If two events A and B are mutually exclusive (disjoint), then P(A or B)=P(A)+P(B).

True, related to the third axiom, disjoint events are additive.

E) The combined probability of all possible outcomes is equal to 1.

True, by the second axiom.

so the correct options are C, D, and E.

If you want to learn more, you can read:

link

Mathematics
Step-by-step answer
P Answered by PhD
I'm assuming you're asked to find \mathbb P(A\mid B), not \mathbb P\left(\dfrac AB\right) (which is nonsense as far as I can tell).

By definition,

\mathbb P(A\mid B)=\dfrac{\mathbb P(A\cap B)}{\mathbb P(B)

We're told the two events are mutually exclusive, which means their intersection is disjoint and the probability of their intersection is 0. So the answer is C.
Mathematics
Step-by-step answer
P Answered by PhD
I'm assuming you're asked to find \mathbb P(A\mid B), not \mathbb P\left(\dfrac AB\right) (which is nonsense as far as I can tell).

By definition,

\mathbb P(A\mid B)=\dfrac{\mathbb P(A\cap B)}{\mathbb P(B)

We're told the two events are mutually exclusive, which means their intersection is disjoint and the probability of their intersection is 0. So the answer is C.
Mathematics
Step-by-step answer
P Answered by PhD

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)=\frac{P(A\cap B)}{P(B)}

P(A\cap B)=P(A/B)\times P(B)

p(A\cap B)=0.26\times 0.5=0.13

P(A)\times P(B)=0.26\times 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\times 0.45=0

Therefore, events B and C are mutually exclusive.

P(A\cap C)=0.26\times 0.26=0.0676

P(A)\times P(C)=0.26\times 0.45=0.117

P(A\cap C)\neq P(A)\cdot P(C)

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

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)

Mathematics
Step-by-step answer
P Answered by PhD

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)=\frac{P(A\cap B)}{P(B)}

P(A\cap B)=P(A/B)\times P(B)

p(A\cap B)=0.26\times 0.5=0.13

P(A)\times P(B)=0.26\times 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\times 0.45=0

Therefore, events B and C are mutually exclusive.

P(A\cap C)=0.26\times 0.26=0.0676

P(A)\times P(C)=0.26\times 0.45=0.117

P(A\cap C)\neq P(A)\cdot P(C)

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

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)

Mathematics
Step-by-step answer
P Answered by PhD

B) 0.55

The probability of the complement of Event A ( P( A⁻ ))  = 0.55

Step-by-step explanation:

Explanation:-

 Given  Event A and Event B are mutually exclusive.

  Given data P(A) = 0.45

                     P(B) = 0.35

                     P(C) = 0.25

The probability of the complement of Event A

                   P( A⁻ ) = 1 - P( A)

                              =  1 - 0.45

                              = 0.55

Final answer:-

The probability of the complement of Event A = 0.55

   

Mathematics
Step-by-step answer
P Answered by PhD

B) 0.55

The probability of the complement of Event A ( P( A⁻ ))  = 0.55

Step-by-step explanation:

Explanation:-

 Given  Event A and Event B are mutually exclusive.

  Given data P(A) = 0.45

                     P(B) = 0.35

                     P(C) = 0.25

The probability of the complement of Event A

                   P( A⁻ ) = 1 - P( A)

                              =  1 - 0.45

                              = 0.55

Final answer:-

The probability of the complement of Event A = 0.55

   

Mathematics
Step-by-step answer
P Answered by PhD
If A and B are mutually independent, then \mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B) and \mathbb P(A\cap B)=0.

If A and C are independent, then \mathbb P(A\cap C)=\mathbb P(A)\cdot\mathbb P(C). Ditto for B and C.

The inclusion/exclusion principle says that

\mathbb P(A\cup B\cup C)=\mathbb P(A)+\mathbb P(B)+\mathbb P(C)-\mathbb P(A\cap B)-\mathbb P(A\cap C)-\mathbb P(B\cap C)+\mathbb P(A\cap B\cap C)

Since A and B are mutually exclusive, their intersection is the empty set. It then follows that A\cap B\cap C also is the empty set, so the last probability is also 0.

Plug in what we know:

\mathbb P(A\cup B\cup C)=0.2+0.4+0.1-0-0.2\cdot0.1-0.4\cdot0.1+0

\implies\mathbb P(A\cup B\cup C)=0.64

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