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Basic Concepts of Probability

1. 

a. \{b, c\}

b. \{a\}

c. \emptyset

3. 

a. H=\{h h h, h h t, h t h, h t t, t h h, t h t, t t h\}, M=\{h h h, h h t, h t h, t h h\}

b. H \cap M=\{h h h, h h t, h t h, t h h\}, H \cup M=H, H^{c}=\{t t t\}

c. P(H \cap M)=4 / 8, P(H \cup M)=7 / 8, P\left(H^{c}\right)=1 / 8

d. Mutually exclusive because they have no elements in common.

5. 

a. B=\{b 1, b 2, b 3, b 4\}, R=\{r 1, r 2, r 3, r 4\}, N=\{b 1, b 2, y 1, y 2, g 1, g 2, r 1, r 2\}

b. B \cap R=\emptyset, B \cup R=\{b 1, b 2, b 3, b 4, r 1, r 2, r 3, r 4\}, B \cap N=\{b 1, b 2\}

R \cup N=\{b 1, b 2, y 1, y 2, g 1, g 2, r 1, r 2, r 3, r 4\}

B^{c}=\{y 1, y 2, y 3, y 4, g 1, g 2, g 3, g 4, r 1, r 2, r 3, r 4\}

(B \cup R)^{c}=\{y 1, y 2, y 3, y 4, g 1, g 2, g 3, g 4\}

c. P(B \cap R)=0, P(B \cup R)=8 / 16, P(B \cap N)=2 / 16, P(R \cup N)=10 / 16,

P\left(B^{c}\right)=12 / 16, P\left((B \cup R)^{c}\right)=8 / 16

d. Not mutually exclusive because they have an element in common.

7. 

a. 0.36

b. 0.78

c. 0.64

d. 0.27

e. 0.87

9. 

a. P(A)=0.38, P(B)=0.62, P(A \cap B)=0

b. P(U)=0.37, P(W)=0.33, P(U \cap W)=0

c. 0.7

d. 0.7

e. A and U are not mutually exclusive because P(A \cap U) is the nonzero number 0.15 . A and V are mutually exclusive because P(A \cap V)=0.

11. 

a. "four or less"

b. "an odd number"

c. "no heads" or "all tails"

d. "a freshman"

13. 

a. "All the children are boys".

Event: \{b b g, b g b, b g g, g b b, g b g, g g b, g g g\}

Complement: \{b b b\}

b. "At least two of the children are girls" or "There are two or three girls".

Event: \{b b b, b b g, b g b, g b b\},

Complement: \{b g g, g b g, g g b, g g g\}

c. "At least one child is a boy".

Event: \{g g g\},

Complement: \{b b b, b b g, b g b, b g g, g b b, g b g, g g b\}

d. "There are either no girls, exactly one girl, or three girls".

Event: \{b g g, g b g, g g b\},

Complement: \{b b b, b b g, b g b, g b b, g g g\}

e. "The first born is a boy".

Event: \{g b b, g b g, g g b, g g g\},

Complement: \{b b b, b b g, b g b, b g g\}

15. 0.47

17. 

a. 0.0023

b. 0.9977

C. 0.0009

d. 0.3014

19. 

a. 920 / 1671

b. 668 / 1671

c. 368 / 1671

d. 1220 / 1671

e. 1003 / 1671

21. 

a. \{h h h\}

b. \{h h t, h t h, h t t, t h h, t h t, t t h, t t t\}

c. \{t t t\}