Question

If P(A): P(Ā) = 4:1 then P(Ā)=​

Answer 1

Answer: - P(Ā) = 1/5.

Detailed solution: -

Given: -

P(A): P(Ā) = 4 : 1

Here,

P is for the probability shown for the given event in the bracket. I.e., P(A) shows the probability of A events.

We need to find the value of P(Ā).

Therefore,

$\frac{P(A)}{P(barA)} = \frac{4}{1}$ [Given, P(A)/P(Ā)]

$\frac{P(A)}{1-P(A)} =\frac{4}{1}$

After cross-multiplication, we will get,

1[P(A)] = 4[1-P(A)]

P(A) = 4 - 4P(A)

P(A) will be written in the left side and the number in the right. I.e., 4P(A) will change its side and as well as sign. The sign will be changes from negative (-) to positive (+) while changing its place from right side of the equal to sign to the left of the equal sign.

I.e.,

P(A) + 4P(A) = 4

5P(A) = 4

Therefore,

$P(A) = \frac{4}{5}$

Now,

As we know,

P(Ā) = 1 - P(A)

Then,

By putting the value of P(A) in P(Ā) = 1 - P(A), we will get,

P(Ā) = 1 - P(A)

$P(barA) = 1 - \frac{4}{5}\\ P(barA) = \frac{5-4}{5}\\ P(barA) = \frac{1}{5}$

Therefore,

P(Ā) = $1/5$

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Answer 2

If P(A) : P(Ā) = 4 : 1 then P(Ā) = 1/5

Given :

P(A) : P(Ā) = 4 : 1

To find :

The value of P(Ā)

Solution :

Step 1 of 2 :

Write down the given equation

Here the given equation is

P(A) : P(Ā) = 4 : 1

Step 2 of 2 :

Find the value of P(Ā)

We know that for any event A

P(A) + P(Ā) = 1

⇒ P(A) = 1 - P(Ā)

Thus we get

P(A) : P(Ā) = 4 : 1

$\displaystyle \sf{ \implies }(1 - P( \overline{A})) : P( \overline{A}) = 4 : 1$

$\displaystyle \sf{ \implies } \frac{1 - P( \overline{A})}{ P( \overline{A})} = \frac{4}{1}$

$\displaystyle \sf{ \implies } 4P( \overline{A}) = 1 - P( \overline{A})$

$\displaystyle \sf{ \implies } 5P( \overline{A}) = 1$

$\displaystyle \sf{ \implies } P( \overline{A}) = \frac{1}{5}$

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