# A high school senior applies for admission to two colleges A and B, and suppose that: P(admitted...

## Question:

A high school senior applies for admission to two colleges A and B, and suppose that: P(admitted at A) = p1, P(rejected by B) = p2, and P(rejected by at least one, A or B) = p3.

(i) Calculate the probability that the student is admitted by at least one college.

(ii) Find the numerical value of the probability in part (i), if p1 = 0.6, p2 = 0.2, and p3 = 0.3.

A course in English composition is taken by 10 freshmen, 15 sophomores, 30 juniors, and 5 seniors. If 10 students are chosen at random, calculate the probability that this group will consist of 2 freshmen, 3 sophomores, 4 juniors, and 1 senior.

## Classical Definition of Probability

Suppose the total number of elementary events in a sample space is {eq}n {/eq}, where {eq}n {/eq} is finite and the elementary events are equally likely, and {eq}n(E) {/eq} of them are favorable to an event {eq}E {/eq}. Then probability of event {eq}E {/eq} is defined as {eq}P(E)=\frac{n(E)}{n} {/eq}.

## Answer and Explanation:

A high school senior applies for admission to two colleges {eq}A {/eq} and {eq}B {/eq}, and let the events {eq}A, B {/eq} denote that he is admitted at {eq}A, B {/eq} respectively.

Then,

{eq}P(\text{admitted at }A) = p_1\implies P(A)=p_1, \\ P(\text{rejected by }B) = p_2\implies P(B^c)=p_2\implies P(B)=1-p_2, \\ P(\text{rejected by at least one, }A\text{ or }B) = p_3\implies P(A^c\cup B^c)=p_2\implies P(A\cap B)=1-p_3. {/eq}

(i) So the probability that the student is admitted by at least one college is,

{eq}P(A\cup B)=P(A)+P(B)-P(A\cap B)=p_1+1-p_2-1+p_3=p_1-p_2+p_3 {/eq}.

(ii) The numerical value of the probability in part (i), if {eq}p_1 = 0.6, p_2 = 0.2, {/eq} and {eq}p_3 = 0.3 {/eq}, is {eq}p_1-p_2+p_3 =0.6-0.2+0.3=0.7 {/eq}.

A course in English composition is taken by 10 freshmen, 15 sophomores, 30 juniors, and 5 seniors.

10 student can be chosen at random in {eq}60\choose 10 {/eq} ways,

If 10 students are chosen at random, the number of cases that this group will consist of 2 freshmen, 3 sophomores, 4 juniors, and 1 senior is {eq}{10\choose 2}{15\choose 3}{30\choose 4}{5\choose 1} {/eq}.

So if 10 students are chosen at random, calculate the probability that this group will consist of 2 freshmen, 3 sophomores, 4 juniors, and 1 senior is {eq}\frac{{10\choose 2}{15\choose 3}{30\choose 4}{5\choose 1}}{60\choose 10}\approx0.0372 {/eq}.