Combinatorics in Calculating Probabilities for a Single Event

Q.

Four persons are chosen at random from a group of $3$ men, $2$ women and $4$ children. The chance that exactly $2$: of them are children, is:

1. $\frac{10}{21}$

2. $\frac{11}{13}$

3. $\frac{13}{25}$

4. $\frac{21}{32}$

Q. Two cards are drawn from a well shuffled pack of $52$ cards. Find the probability that one picked up card is a king card and other card is not a king card.

1. $\frac{{}^4{C}_1\times {}^{48}{C}_1}{{}^{52}{C}_2}$
   
2. $\frac{{}^{48}{C}_2}{{}^{52}{C}_2}$
   
3. $\frac{{}^4{C}_2}{{}^{52}{C}_2}$
   
4. $\frac{{}^4{C}_2}{{}^{48}{C}_2}$
   

Q.

A bag contains $6$ white balls numbered $1$ to $6$ and $4$ red balls numbered $7$ to $10$. Find the probability of getting an odd number ball.

Q. There are seven boys and three girls on a school tennis team. The coach must select four people from this group to participate in county championship. What is the probability that two boys and two girls are chosen for the team?

1. $\frac{3}{10}$
2. $\frac{7}{10}$
3. $\frac{2}{3}$
4. None of the above

Q. A bag contains $4$ red, $3$ green and $5$ blue balls. A ball is drawn at random, what is the probability that it is a blue ball?

1. $\frac{{}^5{C}_1}{{}^{12}{C}_5}$
   
2. $\frac{{}^5{C}_1}{{}^{12}{C}_1}$
   
3. $\frac{1}{{}^{12}{C}_1}$
   
4. $\frac{1}{{}^5{C}_1}$
   

Q. Suppose a number is chosen at random from $1$ to $20$. What is the probability that the chosen number is a prime number?

1. $\frac{{}^9{C}_1}{{}^{20}{C}_1}$
   
2. $\frac{{}^8{C}_1}{{}^{20}{C}_1}$
   
3. $\frac{{}^8{C}_1}{{}^9{C}_1}$
   
4. $\frac{{}^9{C}_1}{{}^{19}{C}_1}$
   

Q. $150$ people line up to buy raffle tickets. Every tenth person who buys a ticket gets a pack of chocolates as a promotional item. What is the probability that a randomly chosen person from the line will receive a pack of chocolates?

1. $\frac{1}{15}$
   
2. $\frac{1}{10}$
   
3. $\frac{1}{150}$
   
4. $\frac{10}{15}$
   

Q. An urn contains $5$ white balls and $3$ red balls. A sample of $5$ balls is selected at random from the urn. What is the probability that sample contains $3$ white balls and $2$ red balls?

1. $\frac{15}{28}$
2. $\frac{9}{28}$
3. $\frac{27}{28}$
4. $\frac{21}{28}$

Q. There are $100$ students in a class. $24$ of the students in the class have brown hair. If a student is called up at random, what is the probability that the student possess brown hair?

1. $\frac{6}{25}$
   
2. $\frac{1}{24}$
   
3. $\frac{1}{100}$
   
4. $\frac{24}{25}$
   

Q. In a lottery, four random numbers from $1-20$ are chosen. A winning ticket matches the four numbers in any order. What is the probability of a winning ticket?

1. $\frac{1}{4845}$
2. $\frac{2}{4845}$
3. $\frac{3}{4845}$
4. $\frac{4}{4845}$