Question

engineering math psychology sociology total male 38 81 55 74 248 female 77 41 24 110 252 total 115 122 79 184 500 a) the probability that a randomly selected person is a male is blank. b) the probability that a randomly selected person is majoring in engineering is blank. c) the probability that a randomly selected person is a female math major is blank. d) the probability that a randomly selected person is a male psychology major is blank. e) the probability that a randomly selected female majors in sociology is blank. f) the probability that a randomly selected male majors in math is blank. g) the probability that a randomly selected math major is a female is blank. h) the probability that a randomly selected person is either a male or a math major is blank. i) the probability that a randomly selected person is either a female or a sociology major is blank.

Explanation:

Step1: Recall Probability Formula

Probability of an event \( E \) is \( P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Total students \( = 500 \).

Step2: Part (a) - Male Probability

Favorable (male) \( = 248 \), Total \( = 500 \).
\( P(\text{Male})=\frac{248}{500}=\frac{62}{125}=0.496 \).

Step3: Part (b) - Engineering Probability

Favorable (engineering) \( = 115 \), Total \( = 500 \).
\( P(\text{Engineering})=\frac{115}{500}=\frac{23}{100}=0.23 \).

Step4: Part (c) - Female Math Major

Favorable (female math) \( = 41 \), Total \( = 500 \).
\( P(\text{Female Math})=\frac{41}{500}=0.082 \).

Step5: Part (d) - Male Psychology Major

Favorable (male psych) \( = 55 \), Total \( = 500 \).
\( P(\text{Male Psych})=\frac{55}{500}=\frac{11}{100}=0.11 \).

Step6: Part (e) - Female Sociology Major

Favorable (female soc) \( = 110 \), Female total \( = 252 \).
\( P(\text{Female Soc}|\text{Female})=\frac{110}{252}=\frac{55}{126}\approx0.4365 \).

Step7: Part (f) - Male Math Major

Favorable (male math) \( = 81 \), Male total \( = 248 \).
\( P(\text{Math}|\text{Male})=\frac{81}{248}\approx0.3266 \).

Step8: Part (g) - Female Math Major (Conditional)

Favorable (female math) \( = 41 \), Math total \( = 122 \).
\( P(\text{Female}|\text{Math})=\frac{41}{122}=\frac{1}{3}\approx0.3361 \).

Step9: Part (h) - Male OR Math Major

Use \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \).
\( P(\text{Male})=\frac{248}{500} \), \( P(\text{Math})=\frac{122}{500} \), \( P(\text{Male and Math})=\frac{81}{500} \).
\( P(\text{Male}\cup\text{Math})=\frac{248 + 122 - 81}{500}=\frac{289}{500}=0.578 \).

Step10: Part (i) - Female OR Sociology Major

\( P(A\cup B)=P(A)+P(B)-P(A\cap B) \).
\( P(\text{Female})=\frac{252}{500} \), \( P(\text{Sociology})=\frac{184}{500} \), \( P(\text{Female and Sociology})=\frac{110}{500} \).
\( P(\text{Female}\cup\text{Sociology})=\frac{252 + 184 - 110}{500}=\frac{326}{500}=\frac{163}{250}=0.652 \).

Answer:

a) \( \frac{62}{125} \) (or \( 0.496 \))
b) \( \frac{23}{100} \) (or \( 0.23 \))
c) \( \frac{41}{500} \) (or \( 0.082 \))
d) \( \frac{11}{100} \) (or \( 0.11 \))
e) \( \frac{55}{126} \) (or \( \approx0.437 \))
f) \( \frac{81}{248} \) (or \( \approx0.327 \))
g) \( \frac{1}{3} \) (or \( \approx0.336 \))
h) \( \frac{289}{500} \) (or \( 0.578 \))
i) \( \frac{163}{250} \) (or \( 0.652 \))