Honors Discrete Chapter 15: Multiplication Rule Worksheet

Honors Discrete Chapter 15: Multiplication Rule Worksheet

15.4 - 15.5Events and Probability WorksheetSOLUTIONS

You may need a separate sheet of paper to complete

1)Consider the random experiment of drawing 1 card from a standard deck of 52 cards. Find the events

  1. E1: The card drawn is an Ace
    {AC, AH, AD, AS}
  2. E2: The cards drawn does not have a number on it

{AC, AH, AD, AS, JC, JH, JD, JS,QC, QH, QD, QS, KC, KH, KD, KS }

  1. E3: The card drawn is not red or black.
    IMPOSSIBLE

2)Consider the random experiment of selecting bit strings of length 4.

  1. E1: Exactly three 0’s = {0001, 0010, 0100, 1000}
  2. E2: The same number of 0’s and 1’s= {0011, 0110, 1100, 1010, 0101, 1001}
  3. E3: Exactly twice as many 0’s as 1’s = IMPOSSIBLE
  4. E4: At most one 0. = {1111, 0111, 1011, 1101, 1110}
  5. E5: At least two 1’s. = {0011, 0110, 1100, 1010, 0101, 1001, 1111, 0111, 1011, 1101, 1110}

3)The sample space S = {σ1, σ2, σ3, σ4, σ5}, and suppose Pr(σ1) = 0.36 and Pr(σ2) = 0. 10.

  1. If σ3, σ4, and σ5 all have the same probability, find Pr(σ3).

Pr(σ1) + Pr(σ2) + Pr(σ3) + Pr(σ4) + Pr(σ5) = 1 and X= Pr(σ3) = Pr(σ4) = Pr(σ5)

.36 + .10 + 3X = 1; X = 0.18;

Pr(σ3) = 0.18

  1. If Pr(σ3) =Pr(σ4) + Pr(σ5), find Pr(σ3).

Pr(σ1) + Pr(σ2) + Pr(σ3) + Pr(σ4) + Pr(σ5) = 1 and X= Pr(σ3) = Pr(σ4) + Pr(σ5)

.36 + .10 + 2X = 1; X = 0.27;

Pr(σ3) = 0.27

4)Consider the sample space S = {σ1, σ2, σ3, σ4}. Find the probability assignment

  1. If all outcomes have the same probability.

Pr(σ1) + Pr(σ2) + Pr(σ3) + Pr(σ4) = 1 and X= Pr(σ1) = Pr(σ2)= Pr(σ3)= Pr(σ4)

4X = 1; X = 0.25;

0.25 = Pr(σ1) = Pr(σ2) = Pr(σ3) = Pr(σ4)

  1. If Pr(σ1) = .28 and all other outcomes are equally possible.

Pr(σ1) + Pr(σ2) + Pr(σ3) + Pr(σ4) = 1 and X= Pr(σ2)= Pr(σ3)= Pr(σ4)

0.28 + 3X = 1; X = 0.24;

Pr(σ1) = 0.28 and Pr(σ2) = Pr(σ3) = Pr(σ4) = 0.24

  1. If 2Pr(σ1) = Pr(σ2) = Pr(σ3)= Pr(σ4).

Pr(σ1) + Pr(σ2) + Pr(σ3) + Pr(σ4) = 1 and X= Pr(σ1) and 2X = Pr(σ2)= Pr(σ3)= Pr(σ4)

7X = 1; X = 1/7;

Pr(σ1) = 1/7and Pr(σ2) = Pr(σ3) = Pr(σ4) = 2/7

5)Eight teams are entered in a soccer tournament. Teams T2, …, T7, T8 have the same probability of winning, T1 is three times as likely to win as all the other teams. Write down the sample space, and find the probability assignment.

T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 = 1and X = T2 = T3 = T4 = T5 = T6 = T7 = T8 and 3X = T1

10X = 1; X = 0.10;

0.l0 = T2 = T3 = T4 = T5 = T6 = T7 = T8 and 0.30 = T1

6)State the complement of each of the following events for each random experiment.

  1. Rolling a die twice.
    E1: Two of a kind
    E1C: Two different numbers
    E2: Two prime numbers:
    E2C:Two Composite Numbers or 1 prime and 1 composite
    E3: Even and Odd number:
    E3C:Both even or both odd
    E4: Sum is even:
    E4C:Even and Odd Number
  2. 5 flips of a coin.
    E1: Exactly 3 Heads
    E1C: 0, 1, 2, 4, or 5 Heads
    E2: At least 4 Heads
    E2C: At most 3 Heads
    E3: At most 2 Heads
    E3C: At least 3 Heads
    E4: At least 1 Tails
    E4C: At most 0 Tails

7) A couple is planning to have 4 children and is concerned about their gender.

  1. How many different 4 children outcomes for boys and girls?24 = 16
  2. What is the probability the couple will have exactly 2 boys?
  3. What is the probability the couple will have at least 1 boy?
  4. What is the probability the couple will have at most 2 girls?

8)Draw 2 card from a standard deck of 52 without replacement

  1. How many different ways can two cards be drawn? 52 * 51 = 2652
  2. What is the probability to draw 2 of a kind?
  3. What is the probability to draw 2 different cards by value?;
    Use Complement same kind:
  4. What is the probability of a queen and king?or
  5. What is the probability of an ace then jack?

9)Draw 2 card from a standard deck of 52 with replacement.

  1. How many different ways can two cards be drawn? 52 * 52 = 2704
  2. What is the probability to draw 2 different cards by value?
  3. What is the probability of 2 different cards by suit?
  4. What is the probability of 2 different cards by color?

10) 12 red marbles, 5 green marbles, and 13 blue marbles are in a bag and each time a marble is chosen it is replaced back in the bag for the next draw.

  1. Find Pr(Red then Blue)
  2. Find Pr(Red then Green)
  3. Find Pr(Green and Blue)
  4. Find Pr(Red then Red)

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