Question

In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for parts a through j. (Note that you cannot find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.)

• Let $F$be the event that a student is female.

• Let $M$be the event that a student is male.

• Let $S$be the event that a student has short hair.

• Let $L$be the event that a student has long hair.

a. The probability that a student does not have long hair.

b. The probability that a student is male or has short hair.

c. The probability that a student is a female and has long hair.

d. The probability that a student is male, given that the student has long hair.

e. The probability that a student has long hair, given that the student is male.

f. Of all the female students, the probability that a student has short hair.

g. Of all students with long hair, the probability that a student is female.

h. The probability that a student is female or has long hair.

i. The probability that a randomly selected student is a male student with short hair.

j. The probability that a student is female.

Short answer

a. $P\left(L^{\text{'}}\right)$

b. $P(M\cup S)$

c. role="math" localid="1650283194438" $P(F\cap L)$

d. $P(M\cap L)$

e. $P(L\cap M)$

f. $P\left(\overline{)F}S\right)$

g. $P\left(\overline{)L}F\right)$

h. $P(F\cup L)$

i. $P(M\cap S)$

j.$P\left(F\right)$

Step-by-step solution

Given information

Let $F$be the event that a student is female.

Let $M$be the event that a student is male.

Let $S$be the event that a student has short hair.

Let $L$be the event that a student has long hair.

Part (a) Step 1: Explanation

The probability that a student does not have long hair means there is no student have long hair in a total number of students. The symbol of probability is given by,

$P\left(L^{\text{'}}\right)=1-P\left(L\right)$

Part (b) Step 1: Explanation

The probability that a student is male or has short hair means there are either male students or short hair students. The symbol of probability is given by,

$P(M\cup S)=P\left(M\right)+P\left(S\right)-P(M\cap S)$

Part (c) Step 1: Explanation

The probability that a student is a female and has long hair means there are female students has long hair. The symbol of probability is given by

$P(F\cap L)=P\left(F\right).P\left(L\right)$

Part (d) Step 1: Explanation

The probability that a student is male, given that the student has long hair means there are male students has long hair. The symbol of probability is given by

$P(M\cap L)=P\left(M\right).P\left(L\right)$

Part (e) Step 1: Explanation

The probability that a student has long hair, given that the student is male means students who have long hair are male students. The symbol of probability is given by

$P(L\cap M)=P\left(L\right).P\left(M\right)$

Part (f) Step 1: Explanation

Of all the female students, the probability that a student has short hair means the student has short hair out of all the female students. The symbol of probability is given by

$P\left(\overline{)F}S\right)=\frac{P(F\cap S)}{P\left(S\right)}$

Part (g) Step 1: Explanation

Of all students with long hair, the probability that a student is female means the student is female among all long hair students. The symbol of probability is given by

$P\left(\overline{)L}F\right)=\frac{P(L\cap F)}{P\left(F\right)}$

Part (h) Step 1: Explanation

The probability that a student is female or has long hair means that the students are either female or long hair students. The symbol of probability is given by

$P(F\cap L)=P\left(F\right).P\left(L\right)$

Part (i) Step 1: Explanation

The probability that a randomly selected student is a male student with short hair means the male student with short hair. The symbol of probability is given by

$P(M\cap S)=P\left(M\right).P\left(S\right)$

Part (j) Step 1: Explanation

The probability that a student is female. The symbol of probability is given by$P\left(F\right)$