Of the 16,701 degrees in mathematics given by U.S. colleges and universities in a recent year, 73% were bachelor's degrees, 21% were master's degrees, and the rest were doctorates. Moreover, women earned 48% of the bachelor's degrees, 42% of the master's degrees, and 29% of the doctorates.

(a) How many of the mathematics degrees given in this year were earned by women? Justify your answer.

(b) Are the events "degree earned by a woman" and "degree was a master's degree" independent? Justify your answer using appropriate probabilities.

(c) If you choose 2 of the 16,701 mathematics degrees at random, what is the probability that at least 1 of the 2 degrees was earned by a woman? Show your work.

Total degrees =$16701$

Total bachelor's degrees P(B)=$73\%=0.73$

Total master's degrees P(M) =$21\%=0.21$

Total doctorates P(D) = $6\%=0.06$

Bachelor's degrees women earned P(BW) = role="math" localid="1652855940510" $48\%=0.48$

Master's degrees earned by women P(MW) = role="math" localid="1652855945773" $42\%=0.42$

Doctorates earned by women P(DW) = role="math" localid="1652855955253" $29\%=0.29$

Probability of women and bachelor's degree P(W AND B) = Bachelor's degrees women earned P(BW) $\times$Total bachelor's degrees P(B)role="math" localid="1652856381883" $=0.48\times 0.73=0.3504$

Probability of women and Masters's degree P(W AND M) = P(MW)$\times$P(M)role="math" localid="1652856364841" $=0.42\times 0.21=0.0882$

Probability of women and Doctorates P(W AND D) = P(DW) $\times$P(D) = $0.29\times 0.06=0.0174$

Now upon adding,

P(W) = $0.3504+0.0882+0.0174=0.4560=45.60\%$

The number of mathematics degrees given in this year was earned by women is

$$\begin{gathered} =16701\times 45.60 \\ =7616 \end{gathered}$$

We know,

P(W) =$45.60\%$and Master's degrees earned by women P(MW) = $42\%$

They are not equal and hence not independent.

We know,

P(W) = $45.60\%$

Let the probability of degrees earned by someone whos not a woman be = P(X)

Hence,

P(W)+P(X)=$1$

= P(X) = $1-$P(W)

role="math" localid="1652857060314" $=1-0.4560=0.5440$

Probability of degrees earned by two people who are not a woman be = $2\times 0.5440=0.295$

The probability that at least one of the two degrees was earned by a woman is

$=1-$$2$P(X)

$$\begin{gathered} =1-0.295 \\ =0.704=70.4\% \end{gathered}$$