72.) given the number cypher wheel at right:
a.) what is the probability of spinning the wheel and landing
on a section that is a composite number and a consonant?
b.) what is the probability of spinning the wheel and landing
on a section that has a prime number and vowel?
c.) what is the probability of spinning the wheel and landing
on a section that has a vowel and a multiple of 3?
d.) what is the probability of spinning the wheel and landing
on a section that has a consonant and a multiple of 5?
e.) what is the probability of spinning the wheel and landing
on a section that has a vowel and a multiple of 2?
f.) what is the probability of spinning the wheel and landing
on a letter of your first name that is also paired with a digit in your street address?
73.) the following table shows music preferences found by a survey of the faculty at a local university
express your answers in fraction form.
| | country music (c) | rock music (r) | oldies (o) | total |
| --- | --- | --- | --- | --- |
| northern u.s. (n) | 11 | 88 | 49 | 148 |
| southern u.s. (s) | 70 | 50 | 44 | 164 |
| total | 81 | 138 | 93 | 312 |
a. find the probability that a randomly selected person from this group likes country music.
b. what is the probability that a randomly selected person from this group likes rock music and
is from the north?
c. find the probability that a randomly selected person from this group likes oldies given that
they are from the south.
d. find ( p(r) ) in decimal form. round to two decimal places.
e. find ( p(s) ) in decimal form. round to two decimal places.
f. find ( p(r|s) ) and explain if events ( r ) and ( s ) are independent or associated events.
g. zoe and lisa are having a disagreement about conditional probability. zoe thinks that
( p(a|b) = p(b|a) ) for any two events ( a ) and ( b ). lisa says that ( p(a|b) ) and ( p(b|a) ) do not have to
be equal. pick two events from the table in problem 5 and determine who is correct.
(see table from problem 5 above)

Question

Accepted Answer

### Part 72a
# Explanation:
## Step1: Identify total sections
The cypher wheel has 26 letters (A - Z) and numbers? Wait, looking at the wheel, each section has a letter and a number. Let's count the sections. From A(0) to Z(25), so 26 sections (since 26 letters, 0 - 25 numbers, 26 sections total).

## Step2: Identify composite numbers and consonants
First, recall: Composite numbers (positive integers >1, not prime, 1 is neither, 0 is neither). Vowels: A, E, I, O, U (5 vowels), so consonants: 26 - 5 = 21.

Numbers: 0 - 25. Composite numbers in 0 - 25: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24 (let's check: 4=2×2, 6=2×3, 8=2×4, 9=3×3, 10=2×5, 12=3×4, 14=2×7, 15=3×5, 16=4×4, 18=2×9, 20=4×5, 21=3×7, 22=2×11, 24=4×6. Also, 0 and 1 are neither, primes: 2,3,5,7,11,13,17,19,23.

Now, letters: Consonants (non - vowels: B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z). So for each section, letter (consonant) and number (composite).

Let's list sections with consonant (B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z) and composite number (4,6,8,9,10,12,14,15,16,18,20,21,22,24).

Letters and their numbers:
- B:1 (prime) → no
- C:2 (prime) → no
- D:3 (prime) → no
- F:5 (prime) → no
- G:6 (composite) → G is consonant, number 6 (composite) → yes
- H:7 (prime) → no
- J:10 (composite) → J is consonant, 10 (composite) → yes
- K:11 (prime) → no
- L:12 (composite) → L is consonant, 12 (composite) → yes
- M:13 (prime) → no
- N:14 (composite) → N is consonant, 14 (composite) → yes
- P:15 (composite) → P is consonant, 15 (composite) → yes
- Q:16 (composite) → Q is consonant, 16 (composite) → yes
- R:17 (prime) → no
- S:18 (composite) → S is consonant, 18 (composite) → yes
- T:19 (prime) → no
- V:20 (composite) → V is consonant, 20 (composite) → yes
- W:21 (composite) → W is consonant, 21 (composite) → yes
- X:22 (composite) → X is consonant, 22 (composite) → yes
- Y:24 (composite) → Y is consonant, 24 (composite) → yes
- Z:25 (25 is composite? 25 = 5×5, yes! Wait, 25 is composite. Z:25 (composite) → Z is consonant, 25 (composite) → yes
- Also, check other consonants: Let's re - check numbers:

Wait, A:0 (vowel), E:4 (vowel? E is vowel, number 4 (composite) → but we need consonant, so E is vowel, exclude. I:8 (vowel, number 8 (composite) → vowel, exclude. O:15? Wait no, O:15? Wait the wheel: Let's assume the numbering is A=0, B=1, C=2, D=3, E=4, F=5, G=6, H=7, I=8, J=9? Wait no, earlier I thought J=10, maybe I made a mistake. Wait, A(0), B(1), C(2), D(3), E(4), F(5), G(6), H(7), I(8), J(9), K(10), L(11), M(12), N(13), O(14), P(15), Q(16), R(17), S(18), T(19), U(20), V(21), W(22), X(23), Y(24), Z(25). Ah! I had the numbering wrong. So:

- A:0 (vowel, 0 - neither)
- B:1 (consonant, 1 - neither)
- C:2 (consonant, 2 - prime)
- D:3 (consonant, 3 - prime)
- E:4 (vowel, 4 - composite)
- F:5 (consonant, 5 - prime)
- G:6 (consonant, 6 - composite)
- H:7 (consonant, 7 - prime)
- I:8 (vowel, 8 - composite)
- J:9 (consonant, 9 - composite)
- K:10 (consonant, 10 - composite)
- L:11 (consonant, 11 - prime)
- M:12 (consonant, 12 - composite)
- N:13 (consonant, 13 - prime)
- O:14 (vowel, 14 - composite)
- P:15 (consonant, 15 - composite)
- Q:16 (consonant, 16 - composite)
- R:17 (consonant, 17 - prime)
- S:18 (consonant, 18 - composite)
- T:19 (consonant, 19 - prime)
- U:20 (vowel, 20 - composite)
- V:21 (consonant, 21 - composite)
- W:22 (consonant, 22 - composite)
- X:23 (consonant, 23 - prime)
- Y:24 (consonant, 24 - composite)
- Z:25 (consonant, 25 - composite)

Now, correct numbering: 0(A),1(B),2(C),3(D),4(E),5(F),6(G),7(H),8(I),9(J),10(K),11(L),12(M),13(N),14(O),15(P),16(Q),17(R),18(S),19(T),20(U),21(V),22(W),23(X),24(Y),25(Z).

Now, composite numbers (4,6,8,9,10,12,14,15,16,18,20,21,22,24,25? Wait 25 is 5×5, composite. 9 is 3×3, composite. 4=2×2, 6=2×3, 8=2×4, 10=2×5, 12=3×4, 14=2×7, 15=3×5, 16=4×4, 18=2×9, 20=4×5, 21=3×7, 22=2×11, 24=4×6, 25=5×5. Primes: 2,3,5,7,11,13,17,19,23. 0 and 1: neither.

Consonants: B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z (21 letters, since vowels A,E,I,O,U:5).

Now, find sections with consonant (letter) and composite number (number):

- B:1 (neither) → no
- C:2 (prime) → no
- D:3 (prime) → no
- F:5 (prime) → no
- G:6 (composite) → consonant (G) and composite (6) → yes
- H:7 (prime) → no
- J:9 (composite) → consonant (J) and composite (9) → yes
- K:10 (composite) → consonant (K) and composite (10) → yes
- L:11 (prime) → no
- M:12 (composite) → consonant (M) and composite (12) → yes
- N:13 (prime) → no
- P:15 (composite) → consonant (P) and composite (15) → yes
- Q:16 (composite) → consonant (Q) and composite (16) → yes
- R:17 (prime) → no
- S:18 (composite) → consonant (S) and composite (18) → yes
- T:19 (prime) → no
- V:21 (composite) → consonant (V) and composite (21) → yes
- W:22 (composite) → consonant (W) and composite (22) → yes
- X:23 (prime) → no
- Y:24 (composite) → consonant (Y) and composite (24) → yes
- Z:25 (composite) → consonant (Z) and composite (25) → yes

Now count these: G(6), J(9), K(10), M(12), P(15), Q(16), S(18), V(21), W(22), Y(24), Z(25). Wait, let's list again:

G:6 (yes)

J:9 (yes)

K:10 (yes)

M:12 (yes)

P:15 (yes)

Q:16 (yes)

S:18 (yes)

V:21 (yes)

W:22 (yes)

Y:24 (yes)

Z:25 (yes)

Wait, that's 11? Wait, did we miss any? Let's check O:14 (vowel, composite) → vowel, so no. E:4 (vowel, composite) → vowel, no. I:8 (vowel, composite) → vowel, no. U:20 (vowel, composite) → vowel, no.

Wait, let's check the numbers again:

Composite numbers in 0 - 25: 4,6,8,9,10,12,14,15,16,18,20,21,22,24,25.

Consonant letters: B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z.

Now, for each consonant letter, check if number is composite:

- B:1 (no)
- C:2 (no)
- D:3 (no)
- F:5 (no)
- G:6 (yes)
- H:7 (no)
- J:9 (yes)
- K:10 (yes)
- L:11 (no)
- M:12 (yes)
- N:13 (no)
- P:15 (yes)
- Q:16 (yes)
- R:17 (no)
- S:18 (yes)
- T:19 (no)
- V:21 (yes)
- W:22 (yes)
- X:23 (no)
- Y:24 (yes)
- Z:25 (yes)

That's G, J, K, M, P, Q, S, V, W, Y, Z → 11 sections? Wait, 4 is E (vowel), 8 is I (vowel), 14 is O (vowel), 20 is U (vowel). So composite numbers with consonant letters: 6(G),9(J),10(K),12(M),15(P),16(Q),18(S),21(V),22(W),24(Y),25(Z) → 11. Wait, 25 is composite, Z is consonant. 9 is composite, J is consonant. 10 is composite, K is consonant. 12 is composite, M is consonant. 15 is composite, P is consonant. 16 is composite, Q is consonant. 18 is composite, S is consonant. 21 is composite, V is consonant. 22 is composite, W is consonant. 24 is composite, Y is consonant. 25 is composite, Z is consonant. And 6(G). So 11? Wait, total sections:26.

Wait, maybe I made a mistake in numbering. Let's count the number of sections: A(0) to Z(25) → 26 sections (26 letters, 0 - 25 numbers, 26 sections).

Now, the number of favorable outcomes (consonant and composite) is 11? Wait, let's re - check the composite numbers and consonant letters:

Composite numbers: 4,6,8,9,10,12,14,15,16,18,20,21,22,24,25 (15 numbers).

Vowel letters with composite numbers: E(4), I(8), O(14), U(20) → 4 sections (vowel + composite).

So consonant + composite = total composite (15) - vowel + composite (4) = 11. Yes, because composite numbers are 15, 4 of them are with vowels (E, I, O, U), so 15 - 4 = 11 with consonants.

Total sections:26.

So probability = favorable / total = 11/26? Wait, no, wait: Wait, 0 is neither, 1 is neither, primes:9 (2,3,5,7,11,13,17,19,23), composite:26 - 2 (0,1) - 9 (primes)=15. Yes, 26 total sections (0 - 25, 26 numbers, 26 letters). So composite numbers:15, vowels:5, so vowel + composite: E(4), I(8), O(14), U(20) → 4 (since A(0) is vowel but 0 is neither, so A is vowel + neither). So consonant + composite:15 - 4 = 11. So probability is 11/26.

# Answer:
$\frac{11}{26}$

### Part 72b
# Explanation:
## Step1: Identify prime numbers and vowels
Prime numbers:2,3,5,7,11,13,17,19,23 (9 primes in 0 - 25). Vowels: A,E,I,O,U (5 vowels).

## Step2: Find sections with vowel (A,E,I,O,U) and prime number (2,3,5,7,11,13,17,19,23)
Letters and their numbers:
- A:0 (neither) → no
- E:4 (composite) → no
- I:8 (composite) → no
- O:14 (composite) → no
- U:20 (composite) → no
- Wait, vowel letters: A(0), E(4), I(8), O(14), U(20). Now, prime numbers:2,3,5,7,11,13,17,19,23.

Check vowel letters (A,E,I,O,U) and their numbers:

- A:0 (neither) → no
- E:4 (composite) → no
- I:8 (composite) → no
- O:14 (composite) → no
- U:20 (composite) → no

Wait, is there any vowel with a prime number? Let's check the numbering again. Wait, maybe I messed up the letter - number mapping. Wait, A(0), B(1), C(2), D(3), E(4), F(5), G(6), H(7), I(8), J(9), K(10), L(11), M(12), N(13), O(14), P(15), Q(16), R(17), S(18), T(19), U(20), V(21), W(22), X(23), Y

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QUESTION IMAGE

Question

Explanation:

Part 72a

Step1: Identify total sections

Step2: Identify composite numbers and consonants

Step1: Identify prime numbers and vowels

Step2: Find sections with vowel (A,E,I,O,U) and prime number (2,3,5,7,11,13,17,19,23)

Answer:

Part 72b