PROBABILITY

PROBABILITY

15.1 EXPERIMENT :

The word experiment means an operation, which can produce well defined outcomes. The are two types of experiment : (i) Deterministic experiment (ii) Probabilistic or Random experiment

(i) Deterministic Experiment : Those experiment which when repeated under identical conditions, produced the same results or outcome are known as deterministic experiment. For example, Physics or Chemistry experiments performed under identical conditions.

(ii) Probabilistic or Random Experiment :- In an experiment, when repeated under identical conditions donot produce the same outcomes every time. For example, in tossing a coin, one is not sure that if a head or tail will be obtained. So it is a random experiment.

Sample space : The set of all possible out comes of a random experiment is called a sample space associated with it and is generally denoted by $S$. For example, When a dice is tossed then $S=1,2,3,4,5,6$.

Even : A subset of sample space associated with a random experiment is called an event. For example, In tossing a dive getting an even no is an event.

Favorable Event : Let $S$ be a sample space associated with a random experiment and $A$ be event associated with the random experiment. The elementary events belonging to $A$ are know as favorable events to the event A. For example, in throwing a pair of dive, $A$ is defined by “Getting 8 as the sum”. Then following elementary events are as out comes : $(2,6),(3,5),(4,4)(5,3)(6,2)$. So, there are 5 elementary events favorable to event $A$.

PROBABILITY

15.2 PROBABILITY :

If there are $n$ elementary events associated with a random experiment and $m$ of them are favorable to an event $A$, then the probability of happening or occurrence of event $A$ is denoted by $P(A)$

Thus, $P(A)=\frac{\text{ Total number of favourable outcomes }}{\text{ Total number of possible outcomes }}=\frac{m}{n}$

And $0\le P(A)\le 1$

If, $P(A)=0$, then $A$ is called impossible event

If, $P(A)=1$, then $A$ is called sure event

$P(A)+P(\overline{A})=1$

Where $P(A)=$ probability of occurrence of $A$.

$P(\overline{A})=$ probability of non - occurrence of $A$.

PROBABILITY

ILLUSTRATIONS :

Ex. 1 A box contains 5 red balls, 4 green balls and 7 white balls. A ball is drawn at random from the box. Find the probability that the ball drawn is (i) white (ii) neither red nor white

PROBABILITY

Sol. Total number of balls in the bag $=5+4+7=16$

$\therefore$ Total number of elementary events $=16$

(i) There are 7 white balls in the bag.

$\therefore$ Favorable number of elementary events $=7$

Hence, $P($ Getting a white ball $)=\frac{\text{ Total No. favourable elementary events }}{\text{ Total No. of elementary events }}=\frac{7}{16}$

(ii) There are 4 balls that are neither red nor white

$\therefore$ Favorable number of elementary events $=4$

Hence, $P($ Getting neither red not white ball $)=\frac{4}{16}=\frac{1}{4}$

PROBABILITY

Ex. 2 All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting

(i) black face card

(ii) a queen

(iii) a black card.

[CBSE - 2007]

PROBABILITY

Sol. After removing three face cards of spades (king, queen, jack) from a deck of 52 playing cards, there are 49 cards left in the pack. Out of these 49 cards one card can be chosen in 49 ways.

$\therefore$ Total number of elementary events $=49$ (i) There are 6 black face cards out of which 3 face cards of spades are already removed. So, out of remaining 3 black face cards one black face card ban be chosen in 3 ways.

$\therefore$ Favorable number of elementary events $=3$

Hence, $P$ (Getting a black face card) $=\frac{3}{49}$

(ii) There are 3 queens in the remaining 49 cards. So, out of these three queens, on queen can be chosen in 3 ways

$\therefore$ Favorable number of elementary events $=3$

Hence $P$ (Getting a queen) $=\frac{3}{49}$

(iii) There are 23 black cards in the remaining 49 cards, So, out to these 23 black card, one black card can be chosen in 23 ways

$\therefore$ Favorable number of elementary events $=23$

Hence, $P$ (Getting a black card) $=\frac{23}{49}$

PROBABILITY

Ex. 3 A die is thrown, Find the probability of

(i) prime number

(ii) multiple of 2 or 3

(iii) a number greater than 3

PROBABILITY

Sol. In a single throw of die any one of six numbers 1,2,3,4,5,6 can be obtained. Therefore, the tome number of elementary events associated with the random experiment of throwing a die is 6.

(i) Let A denote the event “Getting a prime no”. Clearly, event A occurs if any one of 2,3,5 comes as out come.

$\therefore$ Favorable number of elementary events $=3$

Hence, $P$ (Getting a prime no.) $=\frac{3}{6}=\frac{1}{2}$

(ii) An multiple of 2 or 3 is obtained if we obtain one of the numbers 2,3,4,6 as out comes

$\therefore$ Favorable number of elementary events $=4$

Hence, $P$ (Getting multiple of 2 or 3$)=\frac{4}{6}=\frac{2}{3}$

(iii) The event “Getting a number greater than 3” will occur, if we obtain one of number 4,5,6 as an out come.

$\therefore$ Favorable number of out comes $=3$

Hence, required probability $=\frac{3}{6}=\frac{1}{2}$

PROBABILITY

Ex. 4 Two unbiased coins are tossed simultaneously. Find the probability of getting

(i) two heads

(ii) at least one head

(iii) at most one head.

PROBABILITY

Sol. If two unbiased coins are tossed simultaneously, we obtain any one of the following as an out come : $HH,HT,TH,TT$

$\therefore$ Total number of elementary events $=4$

(i) Two heads are obtained if elementary event $HH$ occurs.

$\therefore$ Favorable number of events $=1$

Hence, $P$ (Two heads) $=\frac{1}{4}$

(ii) At least one head is obtained if any one of the following elementary events happen : $HH,HT,TH$

$\therefore$ favorable number of events $=3$

Hence $P($ At least one head $)=\frac{3}{4}$

(iii) If one of the elementary events $HT,TH,TT$ occurs, than at most one head is obtained

$\therefore$ favorable number of events $=3$ Hence, $P($ At most one head $)=\frac{3}{4}$

PROBABILITY

Ex. 5 A box contains 20 balls bearing numbers, 1,2,3,4…..20. A ball is drawn at random from the box. What is the probability that the number of the ball is

(i) an odd number

(ii) divisible by 2 or 3

(iii) prime number

PROBABILITY

Sol. Here, total numbers are 20.

$\therefore$ Total number of elementary events $=20$

(i) The number selected will be odd number, if it is elected from 1,3,5,7,9,11,13,15,17,19

$\therefore$ Favorable number of elementary events $=10$

Hence, $P($ An odd number $)=\frac{10}{20}=\frac{1}{2}$

(ii) Number divisible by 2 or 3 are $2,3,4,6,8,9,10,12,14,15,16,18,20$

$\therefore$ Favorable number of elementary events $=13$

$P($ Number divisible by 2 or 3$)=\frac{13}{20}$

(iii) There are 8 prime number from 1 to 20 i.e., $2,3,5,7,11,13,17,19$

$\therefore$ Favorable number of elementary events $=8$

$P($ prime number $)=\frac{8}{20}=\frac{2}{5}$

PROBABILITY

Ex. 6 A die is drop at random on the rectangular region as shown in figure. What is the probability that it will land inside the circle with diameter $1m$ ?

PROBABILITY

Sol. Area of rectangular region $=3m\times 2m=6m^2$

Area of circle $=\pi r^2$

$=\pi \times (\frac{1}{2}{)}^2$

$=\frac{\pi }{4}{m}^2$

$\therefore$ Probability that die will land inside the circle

$\begin{array}{rl}& =\frac{\pi /4}{6}\\ & =\frac{\pi }{24}\end{array}$

PROBABILITY

DAILY PRACTICE PROBLEMS 15

OBJECTIVE DPP - 15.1

1. If there coins are tossed simultaneously, then the probability of getting at least two heads, is

(A) $\frac{1}{4}$

(B) $\frac{3}{8}$

(C) $\frac{1}{2}$

(D) $\frac{1}{4}$

PROBABILITY

PROBABILITY

2. A bag contains three green marbles four blue marbles, and two orange marbles. If marble is picked at random, then the probability that it is not a orange marble is

(A) $\frac{1}{4}$

(B) $\frac{1}{3}$

(C) $\frac{4}{9}$

(D) $\frac{7}{9}$

PROBABILITY

PROBABILITY

3. A number is selected from number 1 to 27 . The probability that it is prime is

(A) $\frac{2}{3}$

(B) $\frac{1}{6}$

(C) $\frac{1}{3}$

(D) $\frac{2}{9}$

PROBABILITY

PROBABILITY

4. $IF(P(E)=0.05$, then $P($ not $E)=$

(A) -0.05

(B) 0.5

(C) 0.9

(D) 0.95

PROBABILITY

PROBABILITY

5. A bulb is taken out at random from a box of 600 electric bulbs that contains 12 defective bulbs. Then the probability of a non-defective bulb is

(A) 0.02

(B) 0.98

(C) 0.50

(D) None

PROBABILITY

PROBABILITY

SUBJECTIVE DPP - 15.2

1. To dice are thrown simultaneously. Find the probability of getting :

(i) An even number of the sum

(ii) The sum as a prime number

(iii) A total of at least 10

(iv) A multiple of 2 on one dice and a multiple of 3 on the other.

PROBABILITY

Sol. 1.$$ (i) $\frac{1}{2}$ $$ (ii) $\frac{15}{36}$ $$ (iii) $\frac{1}{6}$ $$ (iv) $\frac{11}{36}$

PROBABILITY

2. Find the probability that a leap year selected at random will contain 53 Tuesdays.

PROBABILITY

Sol. 2. $$ $\frac{2}{7}$

PROBABILITY

3. A bag contains 12 balls out of which $x$ are white.

(i) If one ball is drawn at random, what is the probability it will be a white ball ?

(ii) If 6 more white balls are put in the box. The probability of drawing a white ball will be double than that is (i). Find $x$.

PROBABILITY

Sol. 3. $$ (i) $\frac{x}{12}$ $$ (ii) 3

PROBABILITY

4. In a class, there are 18 girls and 16 boys. The class teacher wants to choose one pupil for class monitor. What she does, she writes the name of each pupil a card and puts them into a basket and mixes thoroughly. A child is asked to pick one card from the basket. What is the probability that the name written on the card is : (i) The name of a girl (ii) The name of boy?

PROBABILITY

Sol. 4. $$ (i) $\frac{9}{17}$ $$ (ii) $\frac{8}{17}$

PROBABILITY

5. The probability of selecting a green marble at random from a jar that contains only green, white and yellow marbles is $1/4$. The probability of selecting a white marble from the same jar is $1/3$. If this jar contains 10 yellow marbles. What is the total number of marbles in the jar?

PROBABILITY

Sol. 5. $$ $24$

PROBABILITY

6. A card is drawn at random from a well suffled desk of playing cards. Find the probability that the card drawn is

(i) A card of spade or an ace

(iii) Neither a king nor a queen

(ii) A red king

(iv) Either a king or a queen

PROBABILITY

Sol. 6.$$ (i) $\frac{4}{13}$ $$ (ii) $\frac{1}{26}$ $$ (iii) $\frac{11}{13}$ $$ (iv) $\frac{2}{13}$

PROBABILITY

7. There are 30 cards of same size in a bag on which number 1 to 30 are written. One card is taken out of the bag at random. Find the probability that the number of the selected card is not divisible by 3.

PROBABILITY

Sol. 7. $$ $\frac{2}{3}$

PROBABILITY

8. In figure points $A,B,C$ and $D$ are the centers of four circles that each have a radius of length on unit. If a point is selected at random from the interior of square $ABCD$. What is the probability that the point will be chosen from the shaded region?

PROBABILITY

Sol. 8. $$ $(1-\frac{\pi }{4})$

PROBABILITY

9. A bag contains 5 white balls, 6 red balls, 6 black balls and 8 green balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is

(i) White

(ii) Red or black

(iii) Note green

(iv) Neither white nor black

[CBSE - 2006]

PROBABILITY

Sol. 9. $$ (i) $\frac{1}{5}$ $$ (ii) $\frac{12}{25}$ $$ (iii) $\frac{17}{25}$ $$ (iv) $\frac{14}{25}$

PROBABILITY

10. A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. Find the probability of getting a black ball.

[CBSE - 2008]

PROBABILITY

Sol. 10. $$ $\frac{3}{5}$

PROBABILITY

11. Cards. marked with number 5 to 50 , are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken out card is

(i) a prime number less than 10.

(ii) a number which is a perfect square.

[CBSE - 2008]

PROBABILITY

Sol. 11. $$ (i) $\frac{1}{23}$ $$ (ii) $\frac{5}{46}$