Question

students were asked if their favorite color is blue and if they were born in colorado. some of the results are summarized below in the two - way table.

	blue	not blue	total
not born in co		85
total			216

a student is randomly chosen from this group. what is the probability that the student’s favorite color is blue given that the student was not born in co?

a 54.5%

b 62.9%

c 53.7%

d none of these

e 79%

Explanation:

Step1: Find total not born in CO

First, find the total number of students not born in Colorado. The total number of students is 216, and the total born in CO is 73. So, total not born in CO is \( 216 - 73 = 143 \).

Step2: Find blue not born in CO

Next, find the number of students not born in CO with favorite color blue. The total blue is total (blue born in CO + blue not born in CO). First, find total blue: total born in CO is 73, with 51 blue, so not blue born in CO is \( 73 - 51 = 22 \). Total not blue is \( 22 + 85 = 107 \), so total blue is \( 216 - 107 = 109 \). Then, blue not born in CO is \( 109 - 51 = 58 \).

Step3: Calculate conditional probability

The conditional probability \( P(\text{Blue} | \text{Not born in CO}) = \frac{\text{Blue and Not born in CO}}{\text{Not born in CO}} = \frac{58}{143} \approx 0.4056 \)? Wait, no, wait, let's recalculate. Wait, total born in CO: 73, so not born in CO: 216 - 73 = 143. Not blue: born in CO not blue is 73 - 51 = 22, not born in CO not blue is 85, so total not blue is 22 + 85 = 107. Total blue: 216 - 107 = 109. Blue not born in CO: 109 - 51 = 58. Then \( \frac{58}{143} \approx 0.4056 \)? But that's not matching. Wait, maybe I made a mistake. Wait, the table: Born in CO: Blue=51, Total=73, so Not blue (born in CO) = 73 - 51 = 22. Not born in CO: Not blue=85, so total not born in CO is Blue (not born in CO) + 85. Total students: 216 = 73 + (Blue not born + 85). So Blue not born = 216 - 73 - 85 = 216 - 158 = 58. Then total not born in CO is 58 + 85 = 143. Then probability is 58 / 143 ≈ 0.4056? But the options are 54.5%, 62.9%, 53.7%, etc. Wait, maybe I messed up the table. Wait, the table:

| | Blue | Not blue | Total |
|----------------|------|----------|-------|
| Born in CO | 51 | | 73 |
| Not born in CO | | 85 | |
| Total | | | 216 |

So Born in CO: Not blue = 73 - 51 = 22. Total Not blue = 22 + 85 = 107. Total Blue = 216 - 107 = 109. Not born in CO: Blue = 109 - 51 = 58. Not born in CO: Total = 58 + 85 = 143. Then \( P(\text{Blue} | \text{Not born in CO}) = \frac{58}{143} \approx 0.4056 \)? No, that's 40.56%, but that's not an option. Wait, maybe the table is different. Wait, maybe the total born in CO is 73, so not born in CO is 216 - 73 = 143. Then, blue not born in CO: let's see, maybe I misread the table. Wait, the total for Born in CO is 73, so Not born in CO total is 216 - 73 = 143. Not blue (Not born in CO) is 85, so blue (Not born in CO) is 143 - 85 = 58. Then probability is 58 / 143 ≈ 0.4056? But the options are 54.5, 62.9, 53.7. Wait, maybe the total born in CO is 73, so Not born in CO is 216 - 73 = 143. Wait, maybe the blue not born in CO is 58, and 58 / 109? No, no. Wait, conditional probability is P(Blue | Not CO) = (Blue and Not CO) / (Not CO). So (Blue and Not CO) is 58, Not CO is 143. 58 / 143 ≈ 0.4056, which is 40.56%, but that's not an option. Wait, maybe I made a mistake in the table. Wait, maybe the total born in CO is 73, so Not born in CO is 216 - 73 = 143. Not blue: born in CO not blue is 73 - 51 = 22, not born in CO not blue is 85, so total not blue is 22 + 85 = 107. Total blue is 216 - 107 = 109. Then blue not born in CO is 109 - 51 = 58. Then 58 / 143 ≈ 0.4056. But the options don't have that. Wait, maybe the table was misread. Wait, maybe the "Total" for Born in CO is 73, so Not born in CO total is 216 - 73 = 143. Then blue not born in CO is 143 - 85 = 58. Then 58 / 143 ≈ 0.4056. But the options are 54.5, 62.9, 53.7. Wait, maybe I made a mistake in the total. Wait, maybe the total is 216, born in CO is 73, so not born in CO is 216 - 73 = 143. Then blue not born in CO: let's check the options. Wait, maybe the table is: Born in CO: Blue=51, Total=73, so Not blue (born in CO)=22. Not born in CO: Not blue=85, so total not born in CO= Blue (not born) + 85. Total blue=51 + Blue (not born). Total students=73 + (Blue not born + 85)=216. So Blue not born=216 -73 -85=58. Then probability=58/(58+85)=58/143≈0.4056. But the options are 54.5, 62.9, 53.7. Wait, maybe the total born in CO is 73, so not born in CO is 216 -73=143. Wait, maybe the blue not born in CO is 85? No, no. Wait, maybe the table is:

Wait, the problem says "Some of the results are summarized below in the two way table". Let me re-express the table:

Row 1: Born in CO: Blue=51, Not blue=?, Total=73

Row 2: Not born in CO: Blue=?, Not blue=85, Total=?

Column Total: Blue=?, Not blue=?, Total=216

So first, find Not blue (born in CO)=73 -51=22.

Then, total Not blue=22 +85=107.

Thus, total Blue=216 -107=109.

Then, Blue (not born in CO)=109 -51=58.

Total Not born in CO=58 +85=143.

Now, the conditional probability is P(Blue | Not born in CO)=58/143≈0.4056, which is about 40.56%, but the options don't have that. Wait, maybe I misread the table. Wait, maybe the "Total" for Born in CO is 73, so Not born in CO is 216 -73=143. Then Blue not born in CO: maybe 85? No, 85 is Not blue. Wait, maybe the table is:

| | Blue | Not blue | Total |
|----------------|------|----------|-------|
| Born in CO | 51 | 22 | 73 |
| Not born in CO | x | 85 | y |
| Total | 51+x | 107 | 216 |

So 51 + x + 107 =216 → x=58, y=58+85=143. Then P(Blue|Not born)=58/143≈0.4056. But the options are 54.5, 62.9, 53.7. Wait, maybe the total is 143? No, 216 is total. Wait, maybe the question is "given that the student was born in CO"? No, the question is "not born in CO". Wait, maybe I made a mistake in the total. Wait, maybe the total is 143? No, the table says total is 216. Wait, maybe the options are wrong, or I misread the table. Wait, maybe the "Total" for Born in CO is 73, so Not born in CO is 216 -73=143. Then Blue not born in CO: let's check the options. Wait, 58/143≈0.4056, but the options are 54.5, 62.9, 53.7. Wait, maybe the table is: Born in CO: Blue=51, Total=73, so Not blue=22. Not born in CO: Not blue=85, so total not born in CO=216 -73=143. Then Blue not born in CO=143 -85=58. Then 58/143≈0.4056. But the options are 54.5, 62.9, 53.7. Wait, maybe the total is 109? No, the total is 216. Wait, maybe the question is "given that the student's favorite color is blue, what's the probability they were not born in CO"? No, the question is "favorite color is blue given not born in CO". Wait, maybe I messed up the table. Wait, maybe the "Total" for Not born in CO is 143, and Blue not born in CO is 85? No, 85 is Not blue. Wait, maybe the table is:

| | Blue | Not blue | Total |
|----------------|------|----------|-------|
| Born in CO | 51 | 22 | 73 |
| Not born in CO | 85 |? |? |
| Total |? |? | 216 |

No, that doesn't make sense. Wait, the options include "none of these", but maybe I made a mistake. Wait, let's recalculate:

Wait, maybe the total born in CO is 73, so not born in CO is 216 -73=143.

Blue not born in CO: let's see, the options are 54.5%≈0.545, 62.9%≈0.629, 53.7%≈0.537.

So 0.545*143≈78, 0.629*143≈90, 0.537*143≈77. So maybe Blue not born in CO is 78, 90, or 77.

Wait, maybe the total blue is 51 + 78=129, total not blue=216 -129=87, but not blue born in CO is 22, so not blue not born in CO=87 -22=65, but the table says 85. So that's not matching.

Wait, maybe the table is:

Born in CO: Blue=51, Total=73 → Not blue=22

Not born in CO: Not blue=85, Total=143 → Blue=143 -85=58

Total Blue=51 +58=109, Total Not blue=22 +85=107, Total=216. Correct.

Then probability=58/143≈0.4056≈40.56%, which is not among the options (A:54.5, B:62.9, C:53.7, D:none, E:79). So the answer is D: none of these.

Answer:

D. none of these