Download Honors Discrete Chapter 15: Multiplication Rule Worksheet

15.4 - 15.5 Events and Probability Worksheet SOLUTIONS
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
a. E1: The card drawn is an Ace
{AC, AH, AD, AS}
b. 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 }
c. E3: The card drawn is not red or black.
IMPOSSIBLE
2) Consider the random experiment of selecting bit strings of length 4.
a. E1: Exactly three 0’s = {0001, 0010, 0100, 1000}
b. E2: The same number of 0’s and 1’s = {0011, 0110, 1100, 1010, 0101, 1001}
c. E3: Exactly twice as many 0’s as 1’s = IMPOSSIBLE
d. E4: At most one 0. = {1111, 0111, 1011, 1101, 1110}
e. 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.
a. 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
b. 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
a. 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)
b. 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
c. 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) = 1/7 and 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 = 1 and 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.
a. Rolling a die twice.
b.
5 flips of a coin.
E1: Two of a kind
E1: Exactly 3 Heads
E1C: Two different numbers
E1C: 0, 1, 2, 4, or 5 Heads
E2: Two prime numbers:
E2: At least 4 Heads
E2C: Two Composite Numbers or 1 prime
E2C: At most 3 Heads
and 1 composite
E3: At most 2 Heads
E3C: At least 3 Heads
E3: Even and Odd number:
C
E3 : Both even or both odd
E4: At least 1 Tails
E4C: At most 0 Tails
E4: Sum is even:
C
E4 : Even and Odd Number
7)
A couple is planning to have 4 children and is concerned about their gender.
a. How many different 4 children outcomes for boys and girls? 24 = 16
C
b. What is the probability the couple will have exactly 2 boys?  4 2
16
c. What is the probability the couple will have at least 1 boy? 
d. What is the probability the couple will have at most 2 girls?
4

C1  4 C2  4 C3  4 C4
16
4
C0  4 C1  4 C 2
16
8) Draw 2 card from a standard deck of 52 without replacement
a. How many different ways can two cards be drawn? 52 * 51 = 2652
52  3
b. What is the probability to draw 2 of a kind?
2652
52  48
c. What is the probability to draw 2 different cards by value?
;
2652
52  3 52  48

Use Complement same kind: 1 
52  51
2652
84
44 44
or

2652 2652 2652
44
e. What is the probability of an ace then jack?
2652
d. What is the probability of a queen and king?
9) Draw 2 card from a standard deck of 52 with replacement.
a. How many different ways can two cards be drawn? 52 * 52 = 2704
b. What is the probability to draw 2 different cards by value?
52  48
2704
52  39
2704
52  26
d. What is the probability of 2 different cards by color?
2704
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.
c. Find Pr(Green and Blue)
12 13

a. Find Pr(Red then Blue)
c. What is the probability of 2 different cards by suit?
30 30
b. Find Pr(Red then Green)
12 5

30 30
5 13 13 5
 13 5 


 or 2  
30 30 30 30
 30 30 
12 12

d. Find Pr(Red then Red)
30 30