Supposed there are 3 men who are all owners of their own smarties factories. burr kelly, being the brightest and most innovative of the men, produces considerably more smarties than his competitors and has a commanding 45% of the market share. yousef see, who inherited his riches, lags behind burr and produces 35% of the world’s smarties. finally stan furd, brings up the rear with a measly 20%. however, a recent string of smarties related food poisoning has forced the fda investigate these factories to find the root of the problem. through his investigations, the inspector found that one smarty out of every 100 at kelly’s factory was poisonous. at see’s factory, 1.5% of smarties produced were poisonous. and at furd’s factory, the probability a smarty was poisonous was 0.02.(a) what is the probability that a randomly selected smarty will be safe to eat?
(b) if we know that a certain smarty didn’t come from burr kelly’s factory, what is the probability that this smarty is poisonous? (c) given this information, if a randomly selected smarty is poisonous, what is the probability it came from stan furd’s smarties factory?

(a) the probability that a randomly selected Smarty will be safe to eat is 0.98625

(b)If we know that a certain Smarty didn’t come from Burr Kelly’s factory, the probability that this Smarty is poisonous is 0.01681

(c) if a randomly selected Smarty is poisonous, the probability it came from Stan Furd’s Smarties Factory is 0.2909

Step-by-step explanation:

Let call:

K: the smarty come from Burr Kelly's factory

S: The smarty come from Yousef See's factory

F: The smarty come from Stan Furd's factory

P: the smarty is poisonous

NP: The smarty is not poisonous

so, from the question we have 6 events with their respective probabilities:

1. the smarty come from Burr Kelly's factory and is poisonous:

P(K∩P)=0.45*0.01=0.0045

2. the smarty come from Burr Kelly's factory and is not poisonous:

P(K∩NP)=0.45*0.99=0.4455

3. the smarty come from Yousef See's factory and is poisonous:

P(S∩P)=0.35*0.015=0.00525

4. the smarty come from Yousef See's factory and is not poisonous:

P(S∩NP)=0.35*0.985=0.34475

5. the smarty come from Stan Furd's factory and is poisonous:

P(F∩P)=0.20*0.02=0.004

5. the smarty come from Stan Furd's factory and is not poisonous:

P(F∩NP)=0.20*0.98=0.196

Then for calculate the probability that a randomly selected Smarty will be safe to eat, we need to sum the probability of every event in which is involucrate not poisonous:

P(NP)=P(K∩NP)+P(S∩NP)+P(F∩NP)=0.4455+0.34475+0.196=0.98625

For calculate the probability that this Smarty is poisonous if  we know that a certain Smarty didn’t come from Burr Kelly’s factory, we need to use the following equation:

P(P/K')=P(P∩K')/P(K')

Where K' is the event of select a smarty from Yousef See or Stan Furd's factory. So P(K') and P(P∩K') are calculate as:

P(K')=P(S∩P)+P(S∩NP)+P(F∩P)+P(F∩NP)

P(K')=0.00525+0.34475+0.004+0.196=0.55

P(P∩K')=P(S∩P)+P(F∩P)=0.00525+0.004=0.00925

Replacing on th equation:

P(P/K')=0.00925/0.55=0.01681

For calculate the probability that a smarty came from Stan Furd’s Smarties Factory given that it is poisonous can be calculated as:

P(F/P)=P(F∩P)/P(P)

P(P) can be calculate as:

P(P)=P(K∩P)+P(S∩P)+P(F∩P)=0.0045+0.00525+0.004

P(P)=0.01375

Then replacing values on the equation:

P(F/P)=