Download 15.5 Worksheet Solutions

15.4 - 15.5 Events and Probability Worksheet SOLUTIONS
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 answers to a 4 question true or false test.
a. E1: Exactly three False = {FFFT, FFTF, FTFF, TFFF}
b. E2: The same number of True and False = {FFTT, TFFT, TTFF, TFTF, FTFT, FTTF}
c. E3: Exactly twice as many False as True = IMPOSSIBLE
d. E4: At most one False. = {TTTT, FTTT, TFTT, TTFT, TTTF}
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) = 0.08 and Pr(σ4) = 0.17, find Pr(σ5).
Pr(σ1) + Pr(σ2) + Pr(σ3) + Pr(σ4) + Pr(σ5) = 1
.36 + .10 + .08 + 0.17 + X = 1; X = 0.29
Pr(σ5) = 0.29
c. 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) Write a verbal statement for the COMPLEMENT “Not Statement” of each event.
a. Rolling a die twice.
b.
5 flips of a coin.
E1: Two of a kind
E1: Exactly 3 Heads
C
E1 : Two different numbers
E1C: 0, 1, 2, 4, or 5 Heads
E2: Two prime numbers:
E2C: Two Composite Numbers or 1 prime
and 1 composite
E2: At most 2 Heads
E2C: At least 3 Heads
E3: At least 1 Tails
E3C: At most 0 Tails
E3: Even and Odd number:
E3C: Both even or both odd
6) A couple is planning to have 4 children and is concerned about their gender.
a. How many different 4 children
c. What is the probability the couple will
outcomes for boys and girls?
have at least 1 boy?
24 = 16
b. What is the probability the couple will
have exactly 2 boys?
C2

16
4

4
C1  4 C2  4 C3  4 C4
16
d. What is the probability the couple will
have at most 2 girls?
C0  4 C1  4 C 2

16
4
7) 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.
a. Find Pr(Red then Blue)
d. Find Pr(Green and Blue)
12 13

30 30
b. Find Pr(Red then Green)
12 5

30 30
c. Find Pr(Blue then Blue)
13 13

30 30
5 13 13 5
 13 5 


 or 2  
30 30 30 30
 30 30 
e. Find Pr(Red then Red)
12 12

30 30
f. Find Pr(Red and Green)
5 12 12 5
 12 5 


 or 2  
30 30 30 30
 30 30 
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
b. What is the probability to draw 2 of a kind?
52  3 156
13(4  3)

or
2652 2652
2652
c. What is the probability to draw 2 different cards by value?
52  48 2496

2652
2652
OR Use Complement same kind: 1 
52  3
52  51
d. What is the probability of a queen and king?
84
32

2652 2652 or
44 44

2652 2652
can have either order of queen and king
e. What is the probability of an ace then jack?
44
16

2652 2652
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
c. What is the probability of 2 different cards by suit?
52  39
2704
d. What is the probability of 2 different cards by color?
52  26
2704