Basic Concepts of Probability

Read this section about basic concepts of probability, including spaces, and events. This section discusses set operations using Venn diagrams, including complements, intersections, and unions. Finally, it introduces conditional probability and talks about independent events.

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\}