Basic Concepts of Probability

a. \(\{b, c\}\)

b. \(\{a\}\)

c. \(\emptyset\)

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.

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.

a. \(0.36\)

b. \(0.78\)

c. \(0.64\)

d. \(0.27\)

e. \(0.87\)

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 \(P(A \cap V)=0\).

a. "four or less"

b. "an odd number"

c. "no heads" or "all tails"

d. "a freshman"

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: \(\{b b b, b b g, b g b, b g g\}\)

a. \(0.0023\)

b. \(0.9977\)

C. \(0.0009\)

d. \(0.3014\)

a. \(920 / 1671\)

b. \(668 / 1671\)

c. \(368 / 1671\)

d. \(1220 / 1671\)

e. \(1003 / 1671\)

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. \(\{t t t\}\)