Solutions

    1. {b, c}
    2. {a}
    3. ∅

    1. H = {hhh, hht, hth, htt, thh, tht, tth}, M = {hhh, hht, hth, thh}
    2. H ∩ M = {hhh, hht, hth, thh}, H ∪ M = H, Hc = {ttt}
    3. P(H ∩ M) = 4/8 , P(H ∪ M) = 7∕8, P(Hc)=1∕8
    4. Mutually exclusive because they have no elements in common.

    1. B = {b1, b2, b3, b4}, R = {r1,r2,r3,r4}, N = {b1,b2,y1,y2,g1,g2,r1,r2}
    2. B ∩ R = ∅, B ∪ R = {b1, b2, b3, b4, r1, r2, r3, r4}, B ∩ N = {b1, b2}, R ∪ N = {b1, b2, y1, y2, g1, g2, r1 ,r2, r3, r4}, Bc = {y1, y2, y3, y4, g1, g2, g3, g4, r1, r2, r3, r4}, (B ∪ R)c = {y1, y2, y3, y4, g1, g2, g3, g4}
    3. P(B ∩ R) = 0, P(B ∪ R) = 8∕16, P(B ∩ N) = 2∕16, P( R ∪ N) = 10∕16, P(Bc) = 12∕16, P((B ∪ R)c) = 8/16
    4. Not mutually exclusive because they have an element in common.

    1. 0.36
    2. 0.78
    3. 0.64
    4. 0.27
    5. 0.87

    1. P(A) = 0.38, P(B) = 0.62, P(A ∩ B) = 0
    2. P(U) = 0.37, P(W) = 0.33, P(U ∩ W) = 0
    3. 0.7
    4. 0.7
    5. A and U are not mutually exclusive because P(A ∩ U) is the nonzero number 0.15. A and V are mutually exclusive because P(A ∩ V) = 0.

    1. "four or less"
    2. "an odd number"
    3. "no heads" or "all tails"
    4. "a freshman"

    1. "All the children are boys".

      Event: {bbg, bgb, bgg, gbb, gbg, ggb, ggg},

      Complement: {bbb}

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

      Event: {bbb, bbg, bgb, gbb},

      Complement: {bgg, gbg, ggb, ggg}

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

      Event: {ggg},

      Complement: {bbb, bbg, bgb, bgg, gbb, gbg, ggb}

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

      Event: {bgg, gbg, ggb},

      Complement: {bbb, bbg, bgb, gbb, ggg}

    5. "The firstborn is a boy".

      Event: {gbb, gbg, ggb, ggg},

      Complement: {bbb, bbg, bgb, bgg}

  8. 0.47

    1. 0.0023
    2. 0.9977
    3. 0.0009
    4. 0.3014

    1. 920/1671
    2. 668/1671
    3. 368/1671
    4. 1220/1671
    5. 1003/1671

    1. {hhh}
    2. {hht, hth, htt, thh, tht, tth, ttt}
    3. {ttt}