Mastering Probability: Unlocking the Secrets of Chances

Probability—A Theoretical Approach : Problems is based on tossing a coin(s)

Find the probability of getting a head when a coin is tossed once. Also find the probability of getting a tail.

Harpreet tosses two different coins simultaneously (say, one is of ₹1 and other of ₹2). What is the probability that she gets at least one head?

A game consists of tossing a one-rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails and loses otherwise. Calculate the probability that Hanif will lose the game.

Three different coins are tossed together. Find the probability of getting

1. Exactly two heads.

2. at least two heads

3. at least two tails

Probability—A Theoretical Approach : Problems is based on throwing a die(s)

Suppose we throw a die once.

(i) What is the probability of getting a number greater than 4?

(ii) What is the probability of getting a number less than or equal to 4?

Two dice, one blue and one grey, are thrown at the same time. Write down all the possible outcomes. What is the probability that the sum of the two numbers appearing on the top of the dice is

(i) 8? (ii) 13? (iii) Less than or equal to 12?

A pair of dice is thrown once. Find the probability of getting

1. even number on each dice

2. a total of 9.

Two dice are thrown together. The probability of getting the same number on both dice is

Probability—A Theoretical Approach : Problems is based on playing cards

One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will

(i) be an ace,

(ii) Not be an ace.

A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of getting a black face card.

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is neither a heart nor a king.

The King, Queen and Jack of clubs are removed from a pack of 52 cards and then the remaining cards are well shuffled. A card is selected from the remaining cards. Find the probability of getting a card

1. of spade

2. of black king

3. of club

4. of jacks

Probability—A Theoretical Approach : Problems is based on selection of an object (or) thing from a box(or) bag.

A bag contains a red ball, a blue ball and a yellow ball, all the balls being of the same size. Kritika takes out a ball from the bag without looking into it. What is the probability that she takes out the:

(i) Yellow ball? (ii) Red ball? (iii) Blue ball?

A box contains 3 blue, 2 white, and 4 red marbles. If a marble is drawn at random from the box, what is the probability that it will be

(i) White? (ii) Blue? (iii) Red?

A carton consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. Jimmy, a trader, will only accept the shirts which are good, but Sujatha, another trader, will only reject the shirts which have major defects. One shirt is drawn at random from the carton. What is the probability that

(i) It is acceptable to Jimmy?

(ii) It is acceptable to Sujatha?

A bag contains 18 balls out of which x balls are red. If two more red balls are put in the bag, the probability of drawing a red ball will be times the probability of drawing first case. Find the value of x.

A piggy bank contains hundred 50-p coins, fifty ₹1 coins, twenty ₹2 and ten ₹5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, find the probability the coin falling out will be (i) a 50-p coin, (ii) of value more than ₹1, (iii) of value less than ₹5 (iv) a ₹1 or ₹2 coin.

Probability—A Theoretical Approach : Problems is based on geometry& Spinning wheel

A missing helicopter is reported to have crashed somewhere in the rectangular region shown in Fig. 15.2. What is the probability that it crashed inside the lake shown in the figure?

Figure shows the top view of an open square box that is divided into 6 compartments with walls of equal height. Each of the rectangles D, E, F has twice the area of each of the square A, B and C. when a marble is dropped into the box at random, it falls into one of the compartments. What is the probability that it will fall into compartment F?

A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the numbers 1, 2, 3, ..., 12 as shown in the figure. What is the probability that it will point to

1. 6

2. An even number?

3. A prime number?

4. A number which is a multiple of 5?

A game consists of spinning an arrow which comes to rest pointing at one of the regions (1, 2, or 3) see figure. Are the outcomes 1, 2 and 3 equally likely to occur? Give reason.

Probability: Miscellaneous problems

Two players, Sangeeta and Reshma, play a tennis match. It is known that the probability of Sangeeta winning the match is 0.62. What is the probability of Reshma winning the match?

Savita and Hamida are friends. What is the probability that both will have

(i) Different birthdays?

(ii) The same birthday? (Ignoring a leap year).

What is the probability that a non-leap year has 53 Mondays?

There are 40 students in Class X of a school of whom 25 are girls and 15 are boys. The class teacher has to select one student as a class representative. She writes the name of each student on a separate card, the cards being identical. Then she puts cards in a bag and stirs them thoroughly. She then draws one card from the bag. What is the probability that the name written on the card is the name of

(i) A girl?

(ii) A boy?

In a musical chair game, the person playing the music has been advised to stop playing the music at any time within 2 minutes after she starts playing. What is the probability that the music will stop within the first half-minute after starting?

A number x is selected at random from the numbers 1, 2, 3 and 4. Another number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability that product of x and y is less than 16.

Complete the following statements:

(i) Probability of an event E + Probability of the event ‘not E’ = _________.

(ii) The probability of an event that cannot happen is _________. Such an event is called _________.

(iii) The probability of an event that is certain to happen is ___________. Such an event is called ________.

(iv) The sum of the probabilities of all the elementary events of an experiment is __________.

(v) The probability of an event is greater than or equal to and less than or equal to __________.

Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

(iv) A baby is born. It is a boy or a girl.

Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?

Which of the following cannot be the probability of an event?

(A) 2/3 (B) –1.5 (C) 15% (D) 0.7

If P (E) = 0.05, what is the probability of ‘not E’?

A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out

(i) An orange flavoured candy?

(ii) A lemon flavoured candy?

It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is

(i) Red? (ii) Not red?

A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be

(i) Red? (ii) White? (iii) Not green?

A piggy bank contains hundred 50p coins, fifty ₹1 coins, twenty ₹2 coins and ten ₹5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin

(i) Will be a 50 p coin? (ii) Will not be a ₹5 coin?

Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish (seeFig. 15.4). What is the probability that the fish taken out is a male fish?

A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers

1, 2, 3, 4, 5, 6, 7, 8 (see Fig. 15.5), and these are equally likely outcomes. What is the probability that it will point at

(i) 8?

(ii) An odd number?

(iii) A number greater than 2?

(iv) A number less than 9?

A die is thrown once. Find the probability of getting

(i) A prime number; (ii) a number lying between 2 and 6; (iii) an odd number.

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

(i) a king of red colour (ii) a face card (iii) a red face card

(iv) the jack of hearts (v) a spade (vi) the queen of diamonds

Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random.

(i) What is the probability that the card is the queen?

(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) An ace? (b) A queen?

12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.

(i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot.

What is the probability that this bulb is defective?

(ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears

(i) A two-digit number (ii) A perfect square number (iii) A number divisible by 5.

A child has a die whose six faces show the letters as given below:

The die is thrown once. What is the probability of getting (i) A? (ii) D?

Suppose you drop a die at random on the rectangular region shown in Fig. 15.6. What is the probability that it will land inside the circle with diameter 1m?

A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that

(i) She will buy it? (ii) She will not buy it?

Refer to Example 13. (i) Complete the following table:

(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability. Do you agree with this argument?

Justify your answer.

A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

A die is thrown twice. What is the probability that

(i) 5 will not come up either time? (ii) 5 will come up at least once?

[Hint : Throwing a die twice and throwing two dice simultaneously are treated as the same experiment]

Which of the following arguments are correct and which are not correct? Give reasons for your answer.

(i) If two coins are tossed simultaneously there are three possible outcomes—two heads, two tails or one of each. Therefore, for each of these outcomes, the probability is 1/3.

(ii) If a die is thrown, there are two possible outcomes—an odd number or an even number. Therefore, the probability of getting an odd number is 1/2.

Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on

(i) The same day? (ii) Consecutive days? (iii) Different days?

A die is numbered in such a way that its faces show the numbers 1, 2, 2, 3, 3, 6. It is thrown two times and the total score in two throws is noted. Complete the following table which gives a few values of the total score on the two throws:

What is the probability that the total score is

(i) Even? (ii) 6? (iii) At least 6?

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag.

A box contains 12 balls out of which x are black. If one ball is drawn at random from the box, what is the probability that it will be a black ball? If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find x.

A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3. Find the number of blue balls in the jar.