For the following be sure to give the appropriate probabilit

For the following be sure to give the appropriate probability statements, show work, and give probability values to 4 decimal places where applicable.

a. Estimate the probability that a randomly chosen consumer would have seen advertising for the new product and tried the product.

b. Estimate the probability that at least one of TWO randomly chosen independent consumers would have seen the ad.

c. Given that a randomly chosen consumer has seen the product advertised, estimate the probability that the person has NOT tried the product.

d. Let A be the event that a consumer has tried the product. Let B be the event that a consumer has seen the product advertised. Are A and B independent? Justify your answer with probabilities and explanation.

Solution

A manufacturer of laundry detergent has introduced a new product.

Let us consider , S = Seen the advertidement.

Sn = Not seen the advertisement.

T = Tried the product.

N = Not tried the product.

Here P(tried the product) = P(T) = 1/2

P(not tried the product) = P(N) = 1/2

P( seen add / try ) = P(S/T)= 0.01

P(seen add / not try ) =P(S/N)= 0.25

P(not seen add / try) = P(Sn/T)= 0.05

P(not seen add / not try) = P(Sn/N)=0.60

a) the probability that a randomly chosen consumer would have seen advertising for the new product and tried the product.

P(seen advertising product and tried the product)

These two are dependent event.

P(S and T) = P(S/T) * P(T)

= 0.01 * (1/2)

= 0.005.

This is a probability that a randomly chosen consumer would have seen advertising for the new product and tried the product.

b) the probability that at least one of TWO randomly chosen independent consumers would have seen the ad.

Given that P(Tried the product) = P(T) = 1/2

From a) P(S and T) = 0.05

c) Given that a randomly chosen consumer has seen the product advertised, estimate the probability that the person has NOT tried the product.

That means we have to find the probability that(Not tried the product/Seen the product)

By using Baye\'s theorem,

That is P( N / S) = P(S/N)*P(N) / P(S/N)*P(N)+ P( Sn/N)*P(N)   

= (0.25*0.5) / (0.25*0.5) + (0.60*0.5)

=0.125/0.425 = 0.2941

This is an probability that a randomly chosen consumer has seen the product advertised, estimate the probability that the person has NOT tried the product.