Question

A math class consists of 21 female students and 16 male students. Two students are selected at random to participate in a probability experiment. Compute the following probabilities. Write your answers in decimal form. Round to the nearest thousandth as needed.a. A male is selected, then a female. b. A female is selected, then a male. c. Two males are selected. d. Two females are selected. e. No males are selected.

Answer 1

To solve this problem, we will use the conditional probability definition: Given events A,B such that P(B)>0 we have that

$P(A|B)=\frac{P(\text{A}\cap B)}{P(B)}$

Where P(A|B) means the probability of A given B occurred. This also leads to the following

$P(A|B)\cdot P(B)=P(A\cap B)$

Now, let us define two events. Let M1 be the event that we select a male first and let F2 be the event that we select a female second. We want to calculate the following probabilty

$P(M_1\cap F_2)_{}$

Using the definition of conditional probability, this is the same as

$P(M_1\cap F_2)=P(F_2|M_1)\cdot P(M_1)$

Now, we will calculate P(M1). At the beginning, when we have not picked anyone yet, we have 37 people (21 Female and 16 Male). So the probability of picking a male first is simply

$P(M_1)=\frac{16}{37}$

Now, we will calculate the second probability. Since we already picked one male, we have now 36 people (21 female and 15 male). Then the probability of picking a female is

$P(F_2|M_1)=\frac{21}{36}$

So,

$P(M_1\cap F_2)=\frac{16}{37}\cdot\frac{21}{36}=\frac{28}{111}=0.252$