Combinatorics in Calculating Probabilities for Multiple Events

Q.

A mail-order company business has six telephone lines.

Let X denote the number of lines in use at a specified time.

Suppose the p. m .f of $X$ is as given in the accompanying table.

$x$	0	1	2	3	4	5	6
$\mathrm{P}\left(x\right)$	0.10	0.15	0.20	0.25	0.20	0.05	0.05

Calculate the probability of each of the following events.

(a) {at most three lines are in use}
(b) {fewer than three lines are in use}
(c) {at least three lines are in use}
(d) {between two and five lines, inclusive, are in use}
(e) {between two and four lines, inclusive, are not in use}
(f) {at least four lines are not in use}

Q. $3$ balls are selected at random from a bag containing $4$ red and $5$ blue balls. Find the probability that atleast $1$ red ball is selected.

1. $\frac{{}^4{C}_1\times {}^5{C}_2}{{}^9{C}_3}+\frac{{}^4{C}_2\times {}^5{C}_1}{{}^9{C}_3}$
   $+\frac{{}^4{C}_3}{{}^9{C}_3}$
   
2. $\frac{{}^4{C}_1\times {}^5{C}_2}{{}^9{C}_3}+\frac{{}^4{C}_2\times {}^5{C}_1}{{}^9{C}_3}$
   
3. $\frac{{}^4{C}_1\times {}^5{C}_2}{{}^9{C}_3}+\frac{{}^4{C}_3}{{}^9{C}_3}$
   
4. $\frac{{}^4{C}_2\times {}^5{C}_1}{{}^9{C}_3}+\frac{{}^4{C}_3}{{}^9{C}_3}$
   

Q. A card is drawn at random from a deck of $52$ cards. Find the probability of drawing a heart or spade.

1. $\frac{1}{4}$
   
2. $\frac{1}{2}$
   
3. $\frac{1}{13}$
   
4. $\frac{1}{52}$
   

Q. In an experiment of tossing a coin, if head is obtained then the coin is tossed again and if tail is obtained then a die is rolled. What is the probability of obtaining a tail on the coin and number $3$ on the die?

1. $\frac{1}{8}$
   
2. $\frac{1}{6}$
   
3. $\frac{1}{2}$
   
4. $\frac{1}{12}$
   

Q. A store has six different fitness magazines and three different news magazines. If a customer buys four magazines at random, what is the probability that the customer will buy three fitness magazines and one news mgazine?

1. $\frac{2}{7}$
2. $\frac{3}{7}$
3. $\frac{4}{7}$
4. $\frac{5}{7}$

Q. A child randomly selects $4$ toys from a box containing $2$ bunnies, $5$ dogs and $3$ bears. Find the probability that $2$ dogs and $2$ bears are chosen.

1. $\frac{{}^5{C}_2\times {}^3{C}_2}{{}^{10}{C}_4}$
   
   
2. $\frac{{}^5{C}_2}{{}^{10}{C}_4}$
   
3. $\frac{{}^3{C}_2}{{}^{10}{C}_4}$
   
4. $\frac{{}^5{C}_2+{}^3{C}_2}{{}^{10}{C}_4}$
   

Q.

When a certain coin is flipped, the probability of obtaining a tails is $0.60$. Which of the following is the probability that tails would be obtained exactly $10$ times when the coin is flipped $20$ times?

1. $0.0473$

2. $0.1171$

3. $0.1762$

4. $0.2447$

5. $0.50$

Q. Alexa has an important meeting in the morning. She sets three battery-powered alarm clocks just to be safe. If the first alarm clock has a $\frac{1}{{}^4{C}_1}$ probability of malfunctioning, second alarm clock has a $\frac{1}{{}^3{C}_1}$ probability of malfunctioning and the third alarm clock has a $\frac{1}{{}^5{C}_1}$ probability of malfunctioning, what is the probability that all the three alarm clocks fail at the same time?

1. $\frac{1}{12}$
   
2. $\frac{1}{15}$
   
3. $\frac{1}{20}$
   
4. $\frac{1}{60}$
   

Q. Ian gave each of his $2$ buddies a candy at his birthday celebration, with the treats being $2$ Hazel Nut bars, $2$ Coconut Milk bars, and $1$ Dates bars. What is the probability that both of Ian's friends got the same type of candy?

1. $\frac{1}{5}$
2. $\frac{2}{5}$
3. $\frac{1}{10}$
4. $\frac{4}{5}$

Q. A lucky draw machine is filled with $10$ colored balls as $5$ red, $3$ green, and $2$ blue. Machine selects $2$ balls one by one withouth replacement. What is the probability that no two balls are of the same color?

1. $\frac{31}{45}$
2. $\frac{14}{45}$
3. $\frac{13}{45}$
4. $\frac{32}{45}$