Total Probability or Rule of Elimination

In this class, We discuss Total Probability or Rule of Elimination.

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The reader should have prior knowledge of independent events. Click Here.

We take an example and understand total probability.

Example:

A certain product is manufactured at two plants, p1 and p2.

p1 makes 70% of the requirement, and p2 makes 30%.

p1 meets 90% of the standard products, and p2 meets 80% of the standard products.

How many will meet the standard of 100 items purchased by the user?

Event E: Product is upto standard.

Event E depends on the contribution of p1 and p2 to the market.

When we buy a product from the market, and the product is from p1 or p2.

Total probability depends on both the manufacturer’s p1 and p2.

F1 and F2 are events manufactured at p1, and p2, respectively.

E = (E ∩ F1) U (E ∩ F2)

(E ∩ F1) U (E ∩ F2) are mutually exclusive events.

When we pick a good product from the market, the product is from p1 or p2, not both. so mutually exclusive

P(E) = P(E ∩ F1) + P(E ∩ F2)

P(E) = P(F1) P(E|F1) + P(F2) P(E|F2)

P(E) = 0.7 0.9 + 0.3 0.8

P(E) = 0.87

Out of 100 products 87 are the standard products.

Total Probability:

If an event E can occur only along with event F.

Suppose Event F can occur in n mutually exclusive ways F1 F2 . . Fn.

The Probability of event E P(E) = Σi=1 to n (P(Fi)P(E|Fi))

We use the concept of total probability in Bayes theorem.