Question

a phlebotomist draws the blood of a random sample of 50 patients and determines their blood types as shown.

o a a o o
b o b a o
ab b a b b
o o o b o
ab o a b a
o a a o a
o a o b a
o b a a o
o o o a o
o a o a o

construct a frequency distribution.

blood type | frequency
a | □
ab | □
b | □
o | □

(type whole numbers.)

Explanation:

Step1: Count Blood Type A

Go through the list and count how many times 'A' appears. Let's list the positions:
Row 1: A, A; Row 2: A; Row 3: A; Row 4: A; Row 5: A, A; Row 6: A, A, A; Row 7: A, A; Row 8: A, A; Row 9: A, A; Row 10: A.
Counting: 2 + 1 + 1 + 1 + 2 + 3 + 2 + 2 + 2 + 1 = 17? Wait, better to count each 'A' one by one. Let's do it carefully:

First column (O, B, AB, O, AB, O, O, O, O, O) – no A.
Second column (A, O, B, O, O, A, A, B, O, A) – A: positions 1,6,7,10 → 4
Third column (A, B, A, O, A, A, O, A, O, O) – A: positions 1,3,5,6,8 → 5
Fourth column (O, A, B, B, B, O, B, A, A, A) – A: positions 2,8,9,10 → 4
Fifth column (O, O, B, O, A, A, A, O, O, O) – A: positions 5,6,7 → 3
Total A: 4 + 5 + 4 + 3 = 16? Wait, maybe I made a mistake. Let's count all 'A's:

Looking at the data:

Row 1: O, A, A, O, O → 2 A's
Row 2: B, O, B, A, O → 1 A
Row 3: AB, B, A, B, B → 1 A
Row 4: O, O, O, B, O → 0 A
Row 5: AB, O, A, B, A → 2 A's
Row 6: O, A, A, O, A → 3 A's
Row 7: O, A, O, B, A → 2 A's
Row 8: O, B, A, A, O → 2 A's
Row 9: O, O, A, A, O → 2 A's
Row 10: O, A, O, A, O → 2 A's

Now sum: 2 + 1 + 1 + 0 + 2 + 3 + 2 + 2 + 2 + 2 = 17? Wait, maybe better to list all A's:

Let's write all the blood types in order:

1: O, A, A, O, O

2: B, O, B, A, O

3: AB, B, A, B, B

4: O, O, O, B, O

5: AB, O, A, B, A

6: O, A, A, O, A

7: O, A, O, B, A

8: O, B, A, A, O

9: O, O, A, A, O

10: O, A, O, A, O

Now count A's:

Row 1: 2

Row 2: 1 (fourth element)

Row 3: 1 (third element)

Row 4: 0

Row 5: 2 (third and fifth)

Row 6: 3 (second, third, fifth)

Row 7: 2 (second, fifth)

Row 8: 2 (third, fourth)

Row 9: 2 (third, fourth)

Row 10: 2 (second, fourth)

Now sum: 2 + 1 = 3; 3 + 1 = 4; 4 + 0 = 4; 4 + 2 = 6; 6 + 3 = 9; 9 + 2 = 11; 11 + 2 = 13; 13 + 2 = 15; 15 + 2 = 17. Wait, but let's check again. Maybe I miscounted. Alternatively, let's count each column:

Column 2 (second element of each row):

Row1: A

Row2: O

Row3: B

Row4: O

Row5: O

Row6: A

Row7: A

Row8: B

Row9: O

Row10: A

A's here: row1,6,7,10 → 4

Column 3 (third element):

Row1: A

Row2: B

Row3: A

Row4: O

Row5: A

Row6: A

Row7: O

Row8: A

Row9: A

Row10: O

A's: row1,3,5,6,8,9 → 6

Column 4 (fourth element):

Row1: O

Row2: A

Row3: B

Row4: B

Row5: B

Row6: O

Row7: B

Row8: A

Row9: A

Row10: A

A's: row2,8,9,10 → 4

Column 5 (fifth element):

Row1: O

Row2: O

Row3: B

Row4: O

Row5: A

Row6: A

Row7: A

Row8: O

Row9: O

Row10: O

A's: row5,6,7 → 3

Now total A: 4 (col2) + 6 (col3) + 4 (col4) + 3 (col5) = 17. Wait, but let's check the total number of patients: 10 rows × 5 columns = 50. Let's check other blood types to verify.

Step2: Count Blood Type AB

Looking at the data, AB appears in row3 (first element) and row5 (first element). So row3: AB, row5: AB → 2 times.

Step3: Count Blood Type B

Let's count B's:

Row2: B (first element)

Row3: B (second, fourth, fifth elements) → 3

Row4: B (fourth element)

Row5: B (fourth element)

Row7: B (fourth element)

Row8: B (second element)

Row9: B? No, row9: O, O, A, A, O. Row10: O, A, O, A, O.

Wait, let's list B's:

Row2: first element (B)

Row3: second, fourth, fifth (B, B, B) → 3

Row4: fourth (B)

Row5: fourth (B)

Row7: fourth (B)

Row8: second (B)

Any others?

Row1: no

Row6: no

Row9: no

Row10: no

So count: 1 (row2) + 3 (row3) + 1 (row4) + 1 (row5) + 1 (row7) + 1 (row8) = 8? Wait, let's check again:

Row2: B (1)

Row3: B, B, B (3) → total 4

Row4: B (1) → 5

Row5: B (1) → 6

Row7: B (1) → 7

Row8: B (1) → 8. Yes, 8 B's.

Step4: Count Blood Type O

Total patients: 50. So O = 50 - A - AB - B = 50 - 17 - 2 - 8 = 23. Let's verify by counting O's.

Count O's:

Row1: O, O, O → positions 1,4,5 → 3

Row2: O, O → positions 2,5 → 2 (total 5)

Row3: no O (AB, B, A, B, B) → 0 (5)

Row4: O, O, O, O → positions 1,2,3,5 → 4 (9)

Row5: O → position 2 → 1 (10)

Row6: O, O → positions 1,4 → 2 (12)

Row7: O, O → positions 1,3 → 2 (14)

Row8: O, O → positions 1,5 → 2 (16)

Row9: O, O, O → positions 1,2,5 → 3 (19)

Row10: O, O, O → positions 1,3,5 → 3 (22) Wait, that's 22. But 50 -17 -2 -8 =23. So maybe my count of A, B, AB is wrong.

Wait, let's recount A:

Let's list all A's:

Row1: A, A → 2

Row2: A → 1 (total 3)

Row3: A → 1 (4)

Row5: A, A → 2 (6)

Row6: A, A, A → 3 (9)

Row7: A, A → 2 (11)

Row8: A, A → 2 (13)

Row9: A, A → 2 (15)

Row10: A, A → 2 (17). Yes, 17.

AB: row3 (AB), row5 (AB) → 2. Correct.

B: let's count again:

Row2: B →1

Row3: B, B, B →3 (total 4)

Row4: B →1 (5)

Row5: B →1 (6)

Row7: B →1 (7)

Row8: B →1 (8). Correct.

So O should be 50 -17 -2 -8 =23. Let's count O's again:

Row1: O, O, O → 3 (positions 1,4,5)

Row2: O, O → 2 (positions 2,5) → total 5

Row3: no O → 0 (5)

Row4: O, O, O, O → 4 (positions 1,2,3,5) → 9

Row5: O → 1 (position 2) →10

Row6: O, O → 2 (positions 1,4) →12

Row7: O, O → 2 (positions 1,3) →14

Row8: O, O → 2 (positions 1,5) →16

Row9: O, O, O → 3 (positions 1,2,5) →19

Row10: O, O, O → 3 (positions 1,3,5) →22. Wait, missing 1. Oh, row4: fourth element is B, fifth is O. So row4: O, O, O, B, O → positions 1,2,3,5 → 4. Correct. Row5: AB, O, A, B, A → position 2 is O →1. Row6: O, A, A, O, A → positions 1,4 →2. Row7: O, A, O, B, A → positions 1,3 →2. Row8: O, B, A, A, O → positions 1,5 →2. Row9: O, O, A, A, O → positions 1,2,5 →3. Row10: O, A, O, A, O → positions 1,3,5 →3. Wait, row10: first element O, third O, fifth O →3. So row10: 3. Then total O:

Row1: 3

Row2: 2 (3+2=5)

Row3: 0 (5)

Row4: 4 (5+4=9)

Row5: 1 (9+1=10)

Row6: 2 (10+2=12)

Row7: 2 (12+2=14)

Row8: 2 (14+2=16)

Row9: 3 (16+3=19)

Row10: 3 (19+3=22). Hmm, 22. But 50 -17 -2 -8=23. So maybe I missed an O. Let's check row4: O, O, O, B, O → 4 O's. Correct. Row10: O, A, O, A, O → 3 O's. Correct. Wait, maybe my count of B is wrong. Let's count B's again:

Row2: B (1)

Row3: B, B, B (3) →4

Row4: B (1) →5

Row5: B (1) →6

Row7: B (1) →7

Row8: B (1) →8. Correct. AB: 2. A:17. 17+2+8=27. 50-27=23. So O should be 23. So maybe I missed an O in row4? No, row4 has 4. Row10: 3. Wait, row6: O, A, A, O, A → 2 O's. Row7: O, A, O, B, A → 2 O's. Row8: O, B, A, A, O → 2 O's. Row9: O, O, A, A, O → 3 O's. Row10: 3 O's. Let's add again:

Row1: 3

Row2: 2 (5)

Row3: 0 (5)

Row4: 4 (9)

Row5: 1 (10)

Row6: 2 (12)

Row7: 2 (14)

Row8: 2 (16)

Row9: 3 (19)

Row10: 3 (22). Wait, 22. There's a mistake. Let's list all 50 elements:

Row 1 (5): O, A, A, O, O → elements 1-5: O(1), A(2), A(3), O(4), O(5)

Row 2 (5): B(6), O(7), B(8), A(9), O(10)

Row 3 (5): AB(11), B(12), A(13), B(14), B(15)

Row 4 (5): O(16), O(17), O(18), B(19), O(20)

Row 5 (5): AB(21), O(22), A(23), B(24), A(25)

Row 6 (5): O(26), A(27), A(28), O(29), A(30)

Row 7 (5): O(31), A(32), O(33), B(34), A(35)

Row 8 (5): O(36), B(37), A(38), A(39), O(40)

Row 9 (5): O(41), O(42), A(43), A(44), O(45)

Row 10 (5): O(46), A(47), O(48), A(49), O(50)

Now let's count each blood type:

A: elements 2,3,9,13,23,25,27,28,30,32,35,38,39,43,44,47,49 → let's count:

2,3 (row1) →2

9 (row2) →3

13 (row3) →4

23,25 (row5) →6

27,28,30 (row6) →9

32,35 (row7) →11

38,39 (row8) →13

43,44 (row9) →15

47,49 (row10) →17. Correct, 17 A's.

AB: elements 11,21 →2. Correct.

B: elements 6,8,12,14,15,19,24,34,37 → wait, element 6 (row2, B), 8 (row2, B), 12 (row3, B),14 (row3, B),15 (row3, B),19 (row4, B),24 (row5, B),34 (row7, B),37 (row8, B). Wait, that's 9 B's! Oh! I missed element 8 (row2, B) earlier. So row2: B(6), O(7), B(8), A(9), O(10) → two B's (6 and 8). So my earlier count of B was wrong. Let's recount B:

Row2: B(6), B(8) →2

Row3: B(12), B(14), B(15) →3 (total 5)

Row4: B(19) →1 (6)

Row5: B(24) →1 (7)

Row7: B(34) →1 (8)

Row8: B(37)

Answer:

Step1: Count Blood Type A

Go through the list and count how many times 'A' appears. Let's list the positions:
Row 1: A, A; Row 2: A; Row 3: A; Row 4: A; Row 5: A, A; Row 6: A, A, A; Row 7: A, A; Row 8: A, A; Row 9: A, A; Row 10: A.
Counting: 2 + 1 + 1 + 1 + 2 + 3 + 2 + 2 + 2 + 1 = 17? Wait, better to count each 'A' one by one. Let's do it carefully:

First column (O, B, AB, O, AB, O, O, O, O, O) – no A.
Second column (A, O, B, O, O, A, A, B, O, A) – A: positions 1,6,7,10 → 4
Third column (A, B, A, O, A, A, O, A, O, O) – A: positions 1,3,5,6,8 → 5
Fourth column (O, A, B, B, B, O, B, A, A, A) – A: positions 2,8,9,10 → 4
Fifth column (O, O, B, O, A, A, A, O, O, O) – A: positions 5,6,7 → 3
Total A: 4 + 5 + 4 + 3 = 16? Wait, maybe I made a mistake. Let's count all 'A's:

Looking at the data:

Row 1: O, A, A, O, O → 2 A's
Row 2: B, O, B, A, O → 1 A
Row 3: AB, B, A, B, B → 1 A
Row 4: O, O, O, B, O → 0 A
Row 5: AB, O, A, B, A → 2 A's
Row 6: O, A, A, O, A → 3 A's
Row 7: O, A, O, B, A → 2 A's
Row 8: O, B, A, A, O → 2 A's
Row 9: O, O, A, A, O → 2 A's
Row 10: O, A, O, A, O → 2 A's

Now sum: 2 + 1 + 1 + 0 + 2 + 3 + 2 + 2 + 2 + 2 = 17? Wait, maybe better to list all A's:

Let's write all the blood types in order:

1: O, A, A, O, O

2: B, O, B, A, O

3: AB, B, A, B, B

4: O, O, O, B, O

5: AB, O, A, B, A

6: O, A, A, O, A

7: O, A, O, B, A

8: O, B, A, A, O

9: O, O, A, A, O

10: O, A, O, A, O

Now count A's:

Row 1: 2

Row 2: 1 (fourth element)

Row 3: 1 (third element)

Row 4: 0

Row 5: 2 (third and fifth)

Row 6: 3 (second, third, fifth)

Row 7: 2 (second, fifth)

Row 8: 2 (third, fourth)

Row 9: 2 (third, fourth)

Row 10: 2 (second, fourth)

Now sum: 2 + 1 = 3; 3 + 1 = 4; 4 + 0 = 4; 4 + 2 = 6; 6 + 3 = 9; 9 + 2 = 11; 11 + 2 = 13; 13 + 2 = 15; 15 + 2 = 17. Wait, but let's check again. Maybe I miscounted. Alternatively, let's count each column:

Column 2 (second element of each row):

Row1: A

Row2: O

Row3: B

Row4: O

Row5: O

Row6: A

Row7: A

Row8: B

Row9: O

Row10: A

A's here: row1,6,7,10 → 4

Column 3 (third element):

Row1: A

Row2: B

Row3: A

Row4: O

Row5: A

Row6: A

Row7: O

Row8: A

Row9: A

Row10: O

A's: row1,3,5,6,8,9 → 6

Column 4 (fourth element):

Row1: O

Row2: A

Row3: B

Row4: B

Row5: B

Row6: O

Row7: B

Row8: A

Row9: A

Row10: A

A's: row2,8,9,10 → 4

Column 5 (fifth element):

Row1: O

Row2: O

Row3: B

Row4: O

Row5: A

Row6: A

Row7: A

Row8: O

Row9: O

Row10: O

A's: row5,6,7 → 3

Now total A: 4 (col2) + 6 (col3) + 4 (col4) + 3 (col5) = 17. Wait, but let's check the total number of patients: 10 rows × 5 columns = 50. Let's check other blood types to verify.

Step2: Count Blood Type AB

Looking at the data, AB appears in row3 (first element) and row5 (first element). So row3: AB, row5: AB → 2 times.

Step3: Count Blood Type B

Let's count B's:

Row2: B (first element)

Row3: B (second, fourth, fifth elements) → 3

Row4: B (fourth element)

Row5: B (fourth element)

Row7: B (fourth element)

Row8: B (second element)

Row9: B? No, row9: O, O, A, A, O. Row10: O, A, O, A, O.

Wait, let's list B's:

Row2: first element (B)

Row3: second, fourth, fifth (B, B, B) → 3

Row4: fourth (B)

Row5: fourth (B)

Row7: fourth (B)

Row8: second (B)

Any others?

Row1: no

Row6: no

Row9: no

Row10: no

So count: 1 (row2) + 3 (row3) + 1 (row4) + 1 (row5) + 1 (row7) + 1 (row8) = 8? Wait, let's check again:

Row2: B (1)

Row3: B, B, B (3) → total 4

Row4: B (1) → 5

Row5: B (1) → 6

Row7: B (1) → 7

Row8: B (1) → 8. Yes, 8 B's.

Step4: Count Blood Type O

Total patients: 50. So O = 50 - A - AB - B = 50 - 17 - 2 - 8 = 23. Let's verify by counting O's.

Count O's:

Row1: O, O, O → positions 1,4,5 → 3

Row2: O, O → positions 2,5 → 2 (total 5)

Row3: no O (AB, B, A, B, B) → 0 (5)

Row4: O, O, O, O → positions 1,2,3,5 → 4 (9)

Row5: O → position 2 → 1 (10)

Row6: O, O → positions 1,4 → 2 (12)

Row7: O, O → positions 1,3 → 2 (14)

Row8: O, O → positions 1,5 → 2 (16)

Row9: O, O, O → positions 1,2,5 → 3 (19)

Row10: O, O, O → positions 1,3,5 → 3 (22) Wait, that's 22. But 50 -17 -2 -8 =23. So maybe my count of A, B, AB is wrong.

Wait, let's recount A:

Let's list all A's:

Row1: A, A → 2

Row2: A → 1 (total 3)

Row3: A → 1 (4)

Row5: A, A → 2 (6)

Row6: A, A, A → 3 (9)

Row7: A, A → 2 (11)

Row8: A, A → 2 (13)

Row9: A, A → 2 (15)

Row10: A, A → 2 (17). Yes, 17.

AB: row3 (AB), row5 (AB) → 2. Correct.

B: let's count again:

Row2: B →1

Row3: B, B, B →3 (total 4)

Row4: B →1 (5)

Row5: B →1 (6)

Row7: B →1 (7)

Row8: B →1 (8). Correct.

So O should be 50 -17 -2 -8 =23. Let's count O's again:

Row1: O, O, O → 3 (positions 1,4,5)

Row2: O, O → 2 (positions 2,5) → total 5

Row3: no O → 0 (5)

Row4: O, O, O, O → 4 (positions 1,2,3,5) → 9

Row5: O → 1 (position 2) →10

Row6: O, O → 2 (positions 1,4) →12

Row7: O, O → 2 (positions 1,3) →14

Row8: O, O → 2 (positions 1,5) →16

Row9: O, O, O → 3 (positions 1,2,5) →19

Row10: O, O, O → 3 (positions 1,3,5) →22. Wait, missing 1. Oh, row4: fourth element is B, fifth is O. So row4: O, O, O, B, O → positions 1,2,3,5 → 4. Correct. Row5: AB, O, A, B, A → position 2 is O →1. Row6: O, A, A, O, A → positions 1,4 →2. Row7: O, A, O, B, A → positions 1,3 →2. Row8: O, B, A, A, O → positions 1,5 →2. Row9: O, O, A, A, O → positions 1,2,5 →3. Row10: O, A, O, A, O → positions 1,3,5 →3. Wait, row10: first element O, third O, fifth O →3. So row10: 3. Then total O:

Row1: 3

Row2: 2 (3+2=5)

Row3: 0 (5)

Row4: 4 (5+4=9)

Row5: 1 (9+1=10)

Row6: 2 (10+2=12)

Row7: 2 (12+2=14)

Row8: 2 (14+2=16)

Row9: 3 (16+3=19)

Row10: 3 (19+3=22). Hmm, 22. But 50 -17 -2 -8=23. So maybe I missed an O. Let's check row4: O, O, O, B, O → 4 O's. Correct. Row10: O, A, O, A, O → 3 O's. Correct. Wait, maybe my count of B is wrong. Let's count B's again:

Row2: B (1)

Row3: B, B, B (3) →4

Row4: B (1) →5

Row5: B (1) →6

Row7: B (1) →7

Row8: B (1) →8. Correct. AB: 2. A:17. 17+2+8=27. 50-27=23. So O should be 23. So maybe I missed an O in row4? No, row4 has 4. Row10: 3. Wait, row6: O, A, A, O, A → 2 O's. Row7: O, A, O, B, A → 2 O's. Row8: O, B, A, A, O → 2 O's. Row9: O, O, A, A, O → 3 O's. Row10: 3 O's. Let's add again:

Row1: 3

Row2: 2 (5)

Row3: 0 (5)

Row4: 4 (9)

Row5: 1 (10)

Row6: 2 (12)

Row7: 2 (14)

Row8: 2 (16)

Row9: 3 (19)

Row10: 3 (22). Wait, 22. There's a mistake. Let's list all 50 elements:

Row 1 (5): O, A, A, O, O → elements 1-5: O(1), A(2), A(3), O(4), O(5)

Row 2 (5): B(6), O(7), B(8), A(9), O(10)

Row 3 (5): AB(11), B(12), A(13), B(14), B(15)

Row 4 (5): O(16), O(17), O(18), B(19), O(20)

Row 5 (5): AB(21), O(22), A(23), B(24), A(25)

Row 6 (5): O(26), A(27), A(28), O(29), A(30)

Row 7 (5): O(31), A(32), O(33), B(34), A(35)

Row 8 (5): O(36), B(37), A(38), A(39), O(40)

Row 9 (5): O(41), O(42), A(43), A(44), O(45)

Row 10 (5): O(46), A(47), O(48), A(49), O(50)

Now let's count each blood type:

A: elements 2,3,9,13,23,25,27,28,30,32,35,38,39,43,44,47,49 → let's count:

2,3 (row1) →2

9 (row2) →3

13 (row3) →4

23,25 (row5) →6

27,28,30 (row6) →9

32,35 (row7) →11

38,39 (row8) →13

43,44 (row9) →15

47,49 (row10) →17. Correct, 17 A's.

AB: elements 11,21 →2. Correct.

B: elements 6,8,12,14,15,19,24,34,37 → wait, element 6 (row2, B), 8 (row2, B), 12 (row3, B),14 (row3, B),15 (row3, B),19 (row4, B),24 (row5, B),34 (row7, B),37 (row8, B). Wait, that's 9 B's! Oh! I missed element 8 (row2, B) earlier. So row2: B(6), O(7), B(8), A(9), O(10) → two B's (6 and 8). So my earlier count of B was wrong. Let's recount B:

Row2: B(6), B(8) →2

Row3: B(12), B(14), B(15) →3 (total 5)

Row4: B(19) →1 (6)

Row5: B(24) →1 (7)

Row7: B(34) →1 (8)

Row8: B(37)