A and B are two independent events. The probability that both A and B occur is 1/6and the probability that neither of them occurs is 1/3. The probability of occurrence of A is.a)1/2b)1/6c)5/6d)1/3Correct answer is option 'A'. Can you explain this answer?

Question

A and B are two independent events. The probability that both A and B occur is 1/6and the probability that neither of them occurs is 1/3. The probability of occurrence of A is.&#8203;a)1/2b)1/6c)5/6d)1/3Correct answer is option 'A'. Can you explain this answer?

Answer

The probability that both occur simultaneously is 1/6 and the probability that neither occurs is 1/3
Let P(A)=x, P(B)=y
Then P(A)×P(B) = ⅙ becomes xy = 1/6
And [1−P(A)][1−P(B)]=1/3 becomes (1−x)(1−y)=1/3
On Solving for x and y,
we get x=1/3 or x=1/2 which is the probability of occurrence of A.

Answer

Answer

A/q P(A^B)=P(A)* P(B)=1/6
P(A`^B`)=1-P(A)-P(B)+P(A^B)=1/3
=> P( A) +P(B)=5/6
NOW, PUT P(B)=1/6P(B) AND GET ANS

Answer

Answer

To solve this problem, we can use the concept of probability and the given information. Let's break down the problem into smaller parts and analyze each part individually.

Given information:
- Probability of both A and B occurring: 1/6
- Probability of neither A nor B occurring: 1/3

Let's calculate the probability of A occurring.

Probability of A occurring:
We know that the probability of both A and B occurring is 1/6. Since A and B are independent events, we can multiply their individual probabilities to get the probability of both events occurring together.

Let P(A) be the probability of A occurring, and P(B) be the probability of B occurring.

P(A and B) = P(A) * P(B)

Given that P(A and B) = 1/6, we can rewrite the equation as:

1/6 = P(A) * P(B)

Now, let's consider the probability of neither A nor B occurring.

Probability of neither A nor B occurring:
The probability of neither A nor B occurring is given as 1/3. This means that the complement of the event 'both A and B occurring' is 1/3.

The complement of an event A is denoted as Ac, and its probability is given by:

P(Ac) = 1 - P(A)

Since A and B are independent events, the probability of both not occurring is the product of their individual probabilities.

P(Ac and Bc) = P(Ac) * P(Bc)

Given that P(Ac and Bc) = 1/3, we can rewrite the equation as:

1/3 = P(Ac) * P(Bc)

Now, let's substitute the values we have and solve for P(A).

Solving for P(A):
We have two equations:

1/6 = P(A) * P(B)
1/3 = P(Ac) * P(Bc)

Since A and B are independent events, their complements are also independent events.

P(Ac) = 1 - P(A)
P(Bc) = 1 - P(B)

Substituting these values into the second equation, we get:

1/3 = (1 - P(A)) * (1 - P(B))

Expanding the equation, we have:

1/3 = 1 - P(A) - P(B) + P(A) * P(B)

Rearranging the terms, we get:

1/3 = 1 - P(A) - P(B) + 1/6

Simplifying, we have:

1/3 = 7/6 - P(A) - P(B)

Now, let's substitute the value of P(B) from the first equation:

1/3 = 7/6 - P(A) - 1/6 * P(A)

Multiplying through by 6, we get:

2 = 7 - 6P(A) - P(A)

Combining like terms, we have:

0 = 5 - 7P(A)

Rearranging the equation, we get:

7P(A) = 5

Dividing by 7, we have:

P(A) = 5/7

So, the probability of occurrence of A is 5/7, not 1/2 as mentioned in the options.

Therefore, the correct answer cannot be option 'A'.

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