Question

Brainly Geniuses I need your help
Please Let me get out of this problem ✌

Two groups are competing for the position on boards of directors of a corporation .
the probability that the first group and the second group will win are 0.6 and 0.4 respectively.
father is the first group wins the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins .
find the probability that the new product introduced was bye second group .​

Answer 1

This is the question of Baye's Theorem

Solution ::-)

When there is a competition between the groups for getting position in boards of directors of a corporation.

Let E1 and E2 be the events that first and second group wins respectively.

So,

P(E1) = 0.6

P(E1) = 0.6P(E2) = 0.4

Let A be the event that a new product is introduced.

So,

$P (\dfrac{A}{E_1} ) = 0.7 \\ \\ P( \dfrac{A}{E_2} ) = 0.3$

Hence probability that a new product is introduced by second group

i.e.

$P( \dfrac{E_1}{A} ) = \dfrac{P(E_2).P( \dfrac{A}{E_2}) }{P(E_1).P( \dfrac{A}{E_1}) +P(E_2).P( \dfrac{A}{E_2})} \\ \\ = \frac{0.4 \times 0.3}{0.6 \times 0.7 + 0.4 \times 0.3} \\ \\ = \frac{0.12}{0.42 + 0.12} \\ \\ = \frac{0.12}{0.54} \\ \\ = \frac{12}{54} \\ \\ = \frac{6}{27} \\ \\ = \frac{2}{9}$

Which is the required probability.

Answer 2

Answer:

Given :-

• Two groups are competing for the position on boards of directors of a corporation. The probability that the first group and the second group will win are 0.6 and 0.4 respectively. The first group wins the probability of introduction a new product is 0.7 and the corresponding probability is 0.3 if the second group wins.

To Find :-

• What is the probability that the new product introduce by the second group.

Solution :-

Let, the E₁ be the first group wins.

Then, E₂ be the second group wins

And, the E will be the new product introduce.

Then,

$\longmapsto$ $\sf P(E_1)$ = Probability that the first group wins :

⇒ $\sf 0.6$

➦ $\sf\bold{\dfrac{6}{10}}$

$\longmapsto$ $\sf P(E_2)$ = Probability that the second group wins :

⇒ $\sf 0.4$

➦ $\sf\bold{\dfrac{4}{10}}$

$\longmapsto$ $\sf P(E | E_1)$ = Probability of introduction a new product if the first group wins :

⇒ $\sf 0.7$

➦ $\sf\bold{\dfrac{7}{10}}$

$\longmapsto$ $\sf P(E | E_2)$ = Probability of introduction a new product if the second group wins :

⇒ $\sf 0.3$

➦ $\sf\bold{\dfrac{3}{10}}$

Then, the required probability that the new product is introduced by the second group is P(E₂ | E).

Now, by using Baye's Theorem we get,

↦ $\sf \dfrac{P(E_2) . P(E | E_2)}{P(E_1) . P(E | E_1) + P(E_2) . P(E | E_2)}$

↦ $\sf \dfrac{\dfrac{4}{\cancel{10}} \times \dfrac{3}{\cancel{10}}}{\dfrac{6}{\cancel{10}} \times \dfrac{7}{\cancel{10}} + \dfrac{4}{\cancel{10}} \times \dfrac{3}{\cancel{10}}}$

↦ $\sf \dfrac{4 \times 3}{6 \times 7 + 4 \times 3}$

↦ $\sf \dfrac{12}{42 + 12}$

↦ $\sf \dfrac{\cancel{12}}{\cancel{54}}$

↦ $\sf \dfrac{\cancel{6}}{\cancel{27}}$

➠ $\sf\bold{\red{\dfrac{2}{9}}}$

$\therefore$ The probability that the new product introduce by the second group is $\sf\boxed{\bold{\dfrac{2}{9}}}$.