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# Distribution and One-Way ANOVA

Example

Let event C = taking an English class. Let event D = taking a speech class.

Suppose P(C) = 0.75, P(D) = 0.3, P(C|D) = 0.75 and P(C AND D) = 0.225.

a. Are C and D independent?

b. Are C and D mutually exclusive?

c. What is P(D|C)?

Solution 3.10 a. Yes, because P(C|D) = P(C).

b. No, because P(C AND D) is not equal to zero.

c. P(D|C) = = = 0.3

3.10 A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = 0.40, P(D) = 0.30 and P(B AND D) = 0.20.

a. Find P(B|D).

b. Find P(D|B).

c. Are B and D independent?

d. Are B and D mutually exclusive?

Example

In a box there are three red cards and five blue cards. The red cards are marked with the numbers 1, 2, and 3, and the blue cards are marked with the numbers 1, 2, 3, 4, and 5. The cards are well-shuffled. You reach into the box (you cannot see into it) and draw one card.

Let R = red card is drawn, B = blue card is drawn, E = even-numbered card is drawn.

The sample space S = R1, R2, R3, B1, B2, B3, B4, B5. S has eight outcomes.

• P(R) = . P(B) = . P(R AND B) = 0. (You cannot draw one card that is both red and blue.)

• P(E) = . (There are three even-numbered cards, R2, B2, and B4.)

• P(E|B) = . (There are five blue cards: B1, B2, B3, B4, and B5. Out of the blue cards, there are two even

cards; B2 and B4.)

• P(B|E) = . (There are three even-numbered cards: R2, B2, and B4. Out of the even-numbered cards, to are

blue; B2 and B4.)

• The events R and B are mutually exclusive because P(R AND B) = 0.

• Let G = card with a number greater than 3. G = {B4, B5}. P(G) = . Let H = blue card numbered between

one and four, inclusive. H = {B1, B2, B3, B4}. P(G|H) = . (The only card in H that has a number greater

than three is B4.) Since = , P(G) = P(G|H), which means that G and H are independent.