Question

100 people were asked if they liked math, science, or social studies. everyone answered that they liked at least one.
56 like math
43 like science
35 like social studies
18 like math and science
10 like science and social studies
12 like math and social studies
6 like all three subjects
venn diagram of math, science, social
a) how many people like math only? 20
b) how many people like science only? 9
c) if one person is chosen at random, what is the probability that the person will like science and math? 9/50
d) if one person is chosen at random, what is the probability that the person will like math? 56/100
e) if one person is chosen at random, what is the probability that the person will not like science? type your answer...
f) if one person is chosen at random, what is the probability that the person will like science or social studies? type your answer...
g) if one person is chosen at random, what is the probability that the person will like social studies, but not math? type your answer...

Explanation:

Part E

Step1: Find number not liking science

Total people = 100. People liking science = 43. So not liking science = \(100 - 43 = 57\).

Step2: Calculate probability

Probability = \(\frac{\text{Not liking science}}{\text{Total}}=\frac{57}{100}\).

Step1: Use inclusion - exclusion principle

\(n(S\cup SS)=n(S)+n(SS)-n(S\cap SS)\). \(n(S) = 43\), \(n(SS)=35\), \(n(S\cap SS)=10\). So \(n(S\cup SS)=43 + 35-10=68\).

Step2: Calculate probability

Probability = \(\frac{n(S\cup SS)}{\text{Total}}=\frac{68}{100}=\frac{17}{25}\).

Step1: Find number liking social studies but not math

First, find number liking only social studies and social - science only (but not math). \(n(S\cap SS)-n(\text{all three}) = 10 - 6 = 4\) (science and social studies only). \(n(\text{SS only})=n(SS)-n(S\cap SS)-n(M\cap SS)+n(\text{all three})\) (wait, better way: \(n(SS)=35\), \(n(M\cap SS)=12\), \(n(S\cap SS)=10\), \(n(\text{all three}) = 6\). So \(n(\text{SS only})=35-(12 - 6)-(10 - 6)-6=35 - 6 - 4 - 6 = 19\)? Wait, no. Let's re - calculate. The number of people who like social studies but not math is \(n(SS)-n(M\cap SS)\) (but we have to subtract the part that is also in science). Wait, correct formula: \(n(\text{SS only})=n(SS)-n(M\cap SS)-n(S\cap SS)+n(\text{all three})\). Wait, \(n(M\cap SS)=12\), so the number of people who like social studies and either math or science (but we want only social studies not math). So \(n(\text{SS not M})=n(SS)-n(M\cap SS)\) (but adjust for the three - way intersection). Wait, \(n(M\cap SS)\) includes those who like all three. So the number of people who like social studies and math (but not science) is \(12 - 6 = 6\). The number of people who like social studies and science (but not math) is \(10 - 6 = 4\). So the number of people who like social studies only is \(35-(6 + 4+6)=19\)? Wait, no. Let's use the principle:

Total in social studies: 35.

Number in social studies and math (including all three): 12. So number in social studies and math only (not science) is \(12 - 6 = 6\).

Number in social studies and science (including all three): 10. So number in social studies and science only (not math) is \(10 - 6 = 4\).

Number in all three: 6.

So number in social studies only: \(35-(6 + 4+6)=19\)? Wait, that can't be. Wait, let's start over.

We know:

For math only: 20 (from part A)

Math and science only: \(18 - 6 = 12\)

Math and social studies only: \(12 - 6 = 6\)

All three: 6

Science only: 9 (from part B)

Science and social studies only: \(10 - 6 = 4\)

Now, total people: \(20+12 + 6+6+9+4+\text{social only}=100\)

\(20 + 12+6 + 6+9+4=57\), so social only \(=100 - 57 = 43\)? No, that's wrong. Wait, total in math: 56. Math only (20)+math - science only (12)+math - social only (6)+all three (6)=20 + 12+6+6 = 44. Wait, that's not 56. Oh, I see, the initial part A answer of 20 must be wrong? Wait, no, let's recalculate part A.

To find math only: \(n(M)-n(M\cap S)-n(M\cap SS)+n(M\cap S\cap SS)\)

\(n(M) = 56\), \(n(M\cap S)=18\), \(n(M\cap SS)=12\), \(n(M\cap S\cap SS)=6\)

So math only \(=56-(18 - 6)-(12 - 6)-6=56 - 12 - 6 - 6 = 32\). Oh, the initial part A answer is wrong. But since we have to go with the given? Wait, no, the problem is to solve E, F, G. Let's do it correctly.

Correct way:

For three - set Venn diagram, the formula for the number of elements in only one set:

Only Math: \(n(M)-n(M\cap S)-n(M\cap SS)+n(M\cap S\cap SS)\)

\(n(M) = 56\), \(n(M\cap S)=18\), \(n(M\cap SS)=12\), \(n(M\cap S\cap SS)=6\)

Only Math \(=56-(18)+6-(12)+6=56 - 18 - 12+12=48\)? No, the correct formula is \(n(\text{only }M)=n(M)-n(M\cap S)-n(M\cap SS)+n(M\cap S\cap SS)\)

Wait, \(n(M\cap S)\) includes those who are in all three, so the number of people in M and S but not all three is \(18 - 6 = 12\)

Number of people in M and SS but not all three is \(12 - 6 = 6\)

So only M \(=56-(12 + 6+6)=56 - 24 = 32\)

Only S: \(n(S)-n(M\cap S)-n(S\cap SS)+n(M\cap S\cap SS)\)

\(n(S) = 43\), \(n(M\cap S)=18\), \(n(S\cap SS)=10\), \(n(M\cap S\cap SS)=6\)

Number in S and M but not all three: \(18 - 6 = 12\)

Number in S and SS but not all three: \(10 - 6 = 4\…

Answer:

\(\frac{57}{100}\)

Part F