Lesson 3.3 Addition and Multiplication Rules

Addition Rule

A and B are events. The Addition Rule is:

P(A OR B) = P(A) + P(B) - P(A AND B)

Example: In a box that you cannot see into are three yellow cards numbered 1, 2, 3 and four green cards numbered 1, 2, 3, 4. The experiment is to randomly draw one card.

Let Y = yellow card and G = green card. Each card is equally likely and the sample space is {Y1, Y2, Y3, G1, G2, G3, G4}. There are 7 cards in the sample space.

What is the probability of drawing a green card OR a card with a 1 on it?

There are 4 green cards (G1, G2, G3, G4) and 2 cards with a 1 (Y1, G1). Notice that G1 is repeated.

P(green card) = 4/7

P(card with a 1) = 2/7

P(card that is both green card and has a 1) = 1/7

If G = green card and H = card with a 1, then:

P(G OR H) = P(G) + P(H) - P(G AND H)

P(G OR H) = 4/7 + 2/7 - 1/7 = 5/7

When we add P(G) and P(H), we have counted G1 twice. G AND H refers to card G1. So, when we calculate P(G OR H), we must subtract one occurrence of card G1. That is the reason we must subtract P(G AND H).

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The Addition Rule And Mutually Exclusive

If A and B are mutually exclusive,

then P(A AND B) = 0 and the Addition Rule becomes:

P(A OR B) = P(A) + P(B)

Example: A and B are mutually exclusive events. P(A) = 0.05 and P(B) = 0.62. Find P(A or B).

Since A and B are mutually exclusive, the Addition Rule is P(A OR B) = P(A) + P(B) = 0.05 + 0.62 = 0.67.

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Multiplication Rule

A and B are events. The Multiplication Rule is

P(A AND B) = (P(A))(P(B |A)) or

P(A AND B) = (P(B))(P(A | B))

Example: P(A) = 0.3, P(B) = 0.12, and P(B | A) = 0.2.

Then P(A AND B) = (P(A))(P(B | A)) = (0.3)(0.2) = 0.06.

Now find P(A | B).

Since P(A and B) = (P(B))(P(A | B)), divide both sides of this equation by P(B). Then,

P(A | B) = P(A and B) / P(B) = 0.06 / 0.12 = 0.5.

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The Multiplication Rule and Independence

If A and B are independent, then

P(A | B) = P(A) and P(B | A) = P(B). This means that A does not depend on B and B does not depend on A. The Multiplication Rule becomes:

P(A AND B) = (P(A))(P(B))

Example: Toss 2 fair coins. What is the probability of getting 2 tails?

Since the toss of the second coin does not depend on the toss of the first coin,

P(2 tails) = P(tail AND tail) = (P(tail))(P(tail)) = (1/2)(1/2) = 1/4.

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Other Probability Problems

Example

The example illustrates the addition and multiplication rules, mutually exclusive, and independence.

Please continue to the next section of this lesson.

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