High School

At a bakery, three ovens bake loaves of bread. Oven A produces 25% of the loaves, Oven B produces another 25%, and Oven C produces the remaining 50%. Due to occasional issues, 2% of the loaves from Oven A are undercooked, 4% from Oven B are undercooked, and only 1% from Oven C are undercooked.

If you want to know the probability that a customer's undercooked loaf (event R) was cooked in Oven C, which form of Bayes' rule would you use?

Answer :

To find the probability that a customer's undercooked loaf was baked in Oven C, you should use Bayes' theorem. Bayes' theorem allows us to update the probability estimate for an event based on new evidence. In this case, you want to update the probability for each oven being the source of the undercooked loaf based on the given baking and undercooking rates.

Bayes' theorem formula is:

[tex]P(O_k \mid R) = \frac{P(R \mid O_k) \cdot P(O_k)}{P(R)}[/tex]

where:


  • [tex]P(O_k \mid R)[/tex] is the probability that a loaf is from oven [tex]O_k[/tex] (Oven C in this case) given that it is undercooked.

  • [tex]P(R \mid O_k)[/tex] is the probability of a loaf being undercooked given that it is from oven [tex]O_k[/tex].

  • [tex]P(O_k)[/tex] is the probability of the loaf coming from oven [tex]O_k[/tex].

  • [tex]P(R)[/tex] is the total probability of an undercooked loaf.


Step-by-step calculation:


  1. Identify probabilities:


    • [tex]P(O_C) = 0.50[/tex] (Oven C bakes 50% of the loaves)

    • [tex]P(R \mid O_C) = 0.01[/tex] (1% of loaves from Oven C are undercooked)



  2. Compute the total probability of an undercooked loaf [tex]P(R)[/tex]:

    [tex]P(R) = P(R \mid O_A)\cdot P(O_A) + P(R \mid O_B)\cdot P(O_B) + P(R \mid O_C)\cdot P(O_C)[/tex]

    Where:


    • [tex]P(O_A) = 0.25[/tex], [tex]P(R \mid O_A) = 0.02[/tex]

    • [tex]P(O_B) = 0.25[/tex], [tex]P(R \mid O_B) = 0.04[/tex]


    So:
    [tex]P(R) = 0.02 \cdot 0.25 + 0.04 \cdot 0.25 + 0.01 \cdot 0.50 = 0.005 + 0.01 + 0.005 = 0.02[/tex]


  3. Calculate the probability [tex]P(O_C \mid R)[/tex]:

    [tex]P(O_C \mid R) = \frac{0.01 \cdot 0.50}{0.02} = \frac{0.005}{0.02} = 0.25[/tex]



Therefore, the probability that a customer's undercooked loaf was baked in Oven C is 0.25, or 25%.

In summary, this approach utilizes Bayes' theorem to systematically determine the likelihood of each oven being responsible for an undercooked loaf. By calculating these probabilities, we can make informed assumptions about the source of the undercooked bread in the bakery.

Other Questions