a company has estimated that the probabilities of success for 3 products introduced in the market are 1/5,2/3, and 1/2, respectively. Assuming independence, find the probability that exactly 1 product is successful.

Make a tree diagram assuming the products are introduced one after the other.

Assuming
A=success of product 1, and
a=failure of product 1, etc.

Required probability
=P(Abc)+P(aBc)+P(abC)
=(1/5)(1/3)(1/2)+(4/5)(2/3)(1/2)+(4/5)(1/3)(1/2)
=13/30

To find the probability that exactly 1 product is successful, we can use the binomial probability formula.

The probability of success for the first product is 1/5
The probability of failure for the first product is 1 - 1/5 = 4/5

The probability of success for the second product is 2/3
The probability of failure for the second product is 1 - 2/3 = 1/3

The probability of success for the third product is 1/2
The probability of failure for the third product is 1 - 1/2 = 1/2

Using the binomial probability formula, the probability of exactly 1 success out of 3 trials is:

P(X = 1) = (3 choose 1) * (1/5)^1 * (4/5)^2 * (2/3)^1 * (1/3)^2 * (1/2)^1 * (1/2)^2

Applying the binomial coefficient (3 choose 1) = 3! / (1! * (3-1)!) = 3

P(X = 1) = 3 * (1/5)^1 * (4/5)^2 * (2/3)^1 * (1/3)^2 * (1/2)^1 * (1/2)^2

P(X = 1) = 3 * (1/5) * (4/5)^2 * (2/3) * (1/3)^2 * (1/2)^3

P(X = 1) = 3 * (1/5) * (16/25) * (2/3) * (1/9) * (1/8)

P(X = 1) = (3 * 16 * 2) / (5 * 25 * 3 * 9 * 8)

P(X = 1) = 96 / (11250)

P(X = 1) ≈ 0.008533

Therefore, the probability that exactly 1 product is successful is approximately 0.008533, or 0.8533%.

To find the probability that exactly 1 product is successful, we need to calculate the probability of each product being successful individually and then multiply them together.

Let's denote the probabilities of success for the three products as follows:
P₁ = 1/5 for the first product
P₂ = 2/3 for the second product
P₃ = 1/2 for the third product

Since the products' probabilities of success are assumed to be independent, we can use the multiplication rule for independent events to calculate the probability that exactly one product is successful.

The multiplication rule states that for independent events A and B, the probability of both A and B occurring is equal to the product of their individual probabilities:
P(A and B) = P(A) * P(B)

In this case, we want to find the probability that exactly one product is successful. We can consider three scenarios:

1. The first product is successful, and the other two products are not successful. The probability of this scenario is:
P(success for product 1) * P(failure for product 2) * P(failure for product 3) = P₁ * (1 - P₂) * (1 - P₃)

2. The second product is successful, and the other two products are not successful. The probability of this scenario is:
P(failure for product 1) * P(success for product 2) * P(failure for product 3) = (1 - P₁) * P₂ * (1 - P₃)

3. The third product is successful, and the other two products are not successful. The probability of this scenario is:
P(failure for product 1) * P(failure for product 2) * P(success for product 3) = (1 - P₁) * (1 - P₂) * P₃

Now, we can add up the probabilities of these three scenarios to find the total probability of exactly one product being successful:

P(exactly 1 product successful) = P₁ * (1 - P₂) * (1 - P₃) + (1 - P₁) * P₂ * (1 - P₃) + (1 - P₁) * (1 - P₂) * P₃

Now you can substitute the values of P₁, P₂, and P₃ into the equation to get the numerical answer.