Probability

Probability : It is the numerical measurement of the degree of certainty (or) it is simply how likely something is to happen.

Event and Outcome :

1. Outcome : An Outcome is the final result of an experiment.

ex : picking an ace card from a deck of cards, getting four when we roll a dice.

2. Event : An Event is a set of outcomes.

ex : when we roll dice the probability of getting a number less than five is an event.(This event has four outcomes i.e. 1,2,3,4.)

Note : An Event can have only a single outcome.

Ex : when we roll a dice getting a number less than two has only single outcome i.e. 1.

Experimental Probability (Empirical Probability)

These are based on the results of an actual experiment, so they are called experimental probabilities.

It can be found out by an experiment that is repeated a large number of times.

A trial is when the experiment is performed once.
It is denoted by P(E)
P(E) = No. of trials in which the event happened/Total no. of trials

Example : Ram flipped a coin 20 times, 11 times the outcome is heads and another 9 times the outcome is tails. What are the experimental probabilities of heads and tails ?

Solution : Given, Total no. of trials = 20

No. of trials in which heads is an outcome = 11

No. of trials in which tails is an outcome = 9

P(H) = 11/20 , P(T) = 9/20

Equally Likely Outcomes

If each outcome of an event is as likely to occur as the other, we refer to this by saying that the all possible outcomes are equally likely.

1. Example : Suppose that a bag contains 4 red balls and 1 blue ball, and you draw a ball without looking into the bag. Find whether the following are equally likely or not

(a) Picking a red ball from the bag.

(b) Picking a ball of any color

Solution :

(a) Since there are 4 red balls and only one blue ball, we are more likely to get a red ball than a blue ball. So, the outcomes (a red ball or a blue ball) are not equally likely.

(b) The outcome of drawing a ball of any color from the bag is equally likely.(As any color can be taken, now question becomes a ball out of five balls.)

2. Example : Is Getting heads or tails when we flip a coin equally likely ?

Solution : Yes. Both possible outcomes (heads and tails) has the equal probability of 1/2 , so flipping a coin is equally likely.

Theoretical Probability (Classical Probability)

The Experimental Probability can be found out by repeating an experiment a large no. of times, but repeating an experiment has some limitations, as it may be very expensive or unfeasible or difficult in many situations.

Ex : Repeating the experiment of launching a satellite in order to compute the experimental probability of its failure during launching, The repetition of the phenomenon of an earthquake to compute, the experimental probability of a multi- strayed building getting destroyed in an earthquake...etc.

So, in order to avoid this we assume some assumptions (like we assume in many experiments)such as equally likely outcomes for an experiment which leads to exact i.e. theoretical probability.

It is denoted by P(E)

P(E) = Number of outcomes favorable to E/Number of all possible outcomes of the experiment ,

where we assume that the outcomes of the experiment are equally likely.

Example : Find the probability of getting a head when a coin is tossed once.

Solution : The number of possible outcomes is two — Head (H) and Tail (T) i.e. 2. Let E be the event getting a head’. The number of outcomes favorable to E, (i.e., of getting a head) is 1.

Therefore, P(E) = 1/2

Note : The experimental or empirical probability of an event is based on what has actually happened, while the theoretical probability of the event attempts to predict what will happen on the basis of certain assumptions. As the number of trials in an experiment, go on increasing we may expect the experimental and theoretical probabilities to be nearly the same.

Elementary Event

An event having only one outcome of the experiment is called an elementary event.

Example : Find out whether the following are Elementary events.

1. Flipping a coin.

2. When we roll a dice, Probability of getting a number less than 4

Solution

1. When we flip a coin it has two possible outcomes i.e. heads or tails , the final outcome of the coin will be either heads or tails i.e. only one outcome so flipping a coin is elementary event.

2. Probability of getting a number less than 4 is not an elementary event as it has four different outcomes 1,2,3,4., getting any number from these will satisfy the event.

Sum of probabilities of all the elementary events of an experiment is 1.

Example : Find the sum of probabilities of flipping a coin and say whether it is an elementary event or not ?

Solution : The number of possible outcomes is two — Head (H) and Tail (T) i.e. 2.

Let E be the event ‘getting a head’. The number of outcomes favorable to E, (i.e., of getting a head) is 1.

Let E be the event ‘getting a tail’. The number of outcomes favorable to E, (i.e., of getting a tail) is 1.

Therefore,

P(E) = 1/2

P(E) + P(E) = 1/2 + 1/2

Therefore, sum of probabilities of flipping a coin is 1 and it is an elementary event.

Impossible and Sure Events

Impossible event : An event which is impossible to occur or 0% chance of occurring is called Impossible event .Its Probability is 0 i.e. P(E) = 0.

Sure or Certain Event : An event which is sure to occur or 100% chance of occurring is called Sure Event.

Example : Find out whether the following are impossible or sure events.

1. When we roll a dice, getting a number greater than 6

2. When we roll a dice, getting a number less than 7

Solution :

1. An Impossible event, as there are no favorable outcomes to the event i.e. numbers greater than 6 are not present on dice.

2. A Sure event, as all possible outcomes 1,2,3,4,5,6 are less than 7 and are favorable outcomes of the event.

Note : From the definition of the probability P(E), we see that the numerator (number of outcomes favorable to the event E) is always less than or equal to the denominator (the number of all possible outcomes). Therefore,

0 ≤ P(E) ≤ 1

It is also known as Range of Probability.

Geometric probability

Geometric probability is the calculation of the likelihood that one will hit a particular area of a figure.

It is calculated by dividing the desired area by the total area. In the case of Geometrical probability, there are infinite outcomes.

Geometric probability = Area of the desired region / Area of the total region

Example : A circular region is enclosed by a rectangular region. The dimensions of rectangle are 3m and 2m respectively. The circle is of radius 1m. Find the probability that a die thrown in the rectangular region will land inside the circle ?

Solution : Area of the desired region i.e. circular region = π ∗ (1)² = πm²

Area of the total region i.e. rectangular region = 3 ∗ 2 = 6m²

P(E) = π/6

Complementary Events

Two events are said to be complementary, when one event occur if and only if the another event does not. The sum of probabilities of complementary events is 1 all the time.

Example : Find out whether the following events are complementary ?

1. When flip a coin, getting heads and getting tails.

2. when we roll a dice, getting a number 5 or greater and a number 4 or less

Solution :

1. Complementary, because an outcome of flipping a coin is heads if and only if it is not tails.

2. Complementary event, because a roll is 5 or greater if and only if it is not 4 or less.

P(E) + P(E) = 1

P(E) = 1 − P(E) ,

where, event E representing ’not E’ is called the complement of E. E and E are complementary events.

Playing Cards

A deck of playing Cards consists of 52 Cards which are divided into 4 sets of 13 cards each. Two sets are red colored (diamond and heart) and another two sets are black colored.(spades and clubs)

Black

– Spade

– Club

Red

– Diamond

– Heart

The cards in each set are Ace(A), 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack(J), Queen(Q), King(K).

Kings, queens and jacks are called face cards.

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

1. be a king

2. a red colored card

Solution :

1. There are 4 kings in a deck. Let E be the event ‘the card is a king’.

The number of outcomes favorable to E = 4

The number of possible outcomes = 52

P(E) = 4/52 = 1/13

2. There are 26 red colored cards in a deck. Let E be the event ‘the card is a red colored’.

The number of outcomes favorable to E = 26

The number of possible outcomes = 52

P(E) = 26/52 = 1/2

Textbook Exercise Questions

1. Ex 15.1 Q7.

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 ?

Solution : Let E be the event students not having the same birthday then, not E becomes the event students having same birthday.

Given, P(E) = 0.992 , P(E) = ?

We know that, P(E) + P(E) = 1

P(E) = 1 - P(E) = 1 - 0.992 = 0.008

2. Ex 15.1 Q9

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

(a) red ?

(b) white ?

(c) not green ?

Solution : No. of red marbles = 5

No. of white marbles = 8

No. of green marbles = 4

Total no. of marbles = 5 + 8 + 4 = 17

(a) P(R) = 8/17

(b) P(W) = 4/17

3. Ex 15.1 Q15

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.

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

(b) If the queen is drawn and put aside, what is the probability that the second card picked up is

I. an ace ?

ii. a queen ?

Solution : Total no. of possible outcomes = 5 (As 5 cards are there)

(a) No. of favorable outcomes = 1 (only 1 queen is there )

P(Q) = 1/5

(b) As now queen is put aside, the total no. of possible outcomes = 4

I. P(A) = 1/4

ii. P(Q) = 0/4 (as no queen is left in the pack)

Previous Year Questions

1. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is

(a) a card of spade or an ace.

(b) a black king.

(d) either a king or a queen.

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

(a) exactly two heads

(b) at least two heads

3. A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

4. Two different dice are tossed together. Find the probability :

(a) of getting a doublet

(b) of getting a sum 10, of the numbers on the two dice.

5. A die is thrown once. Find the probability of getting

(a) a number between 2 and 6,

(b) a prime number.

Creative Questions

1. 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

(a) 8 ?

(b) 13 ?

2. 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?

3. A missing helicopter is reported to have crashed somewhere in the rectangular region of dimensions 9 and 4.5 km. What is the probability that it crashed inside the lake which is in the rectangular region of dimensions 3 and 2.5 km?

4. A die is thrown twice. What is the probability that

(a) 5 will not come up either time?

(b) 5 will come up at least once?

5. Two customers Sham 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

(a) the same day ?

(b) consecutive days ?

Multiple Choice Questions(MCQs)

Part 1

1. The probability of event equal to one is called;

(a) Unsure event

(b) Sure Event

(d) Independent event

Ans : (b) Sure event

2. The sum of the probabilities of all the elementary events of an experiment is

(a) 0.5

(b) 1

(d) Can be any real number

Ans : (b) 1

3. The probability that cannot exist among the following:

(a) 0.7

(b) -1.5

(d) None of these

Ans : (b) -1.5

4. If an event cannot occur, then its probability is

(a) 1

(b) a negative number

(d) 0

Ans : (d) 0

5. If P(A) denotes the probability of an event A, then

(a) P(A) < 0

(b) P(A) > 1

(d) –1 ≤ P(A) ≤ 1

Ans : (c) 0 ≤ P(A) ≤ 1

Part 2

1. A number x is chosen at random from the numbers -3,-2,-1,0,1,2,3 the probability that —x— < 2

(a) 5/7

(b) 2/7

(d) 1/7

Ans : (c) 3/7

2. If a number x is chosen from the numbers 1,2,3, and a number y is selected from the numbers 1,4,9.

Then, P(xy > 9)

(a) 5/9

(b) 4/9

(d) 1/9

Ans : (b) 4/9

3. The probability that a non-leap year has 53 Sundays, is

(a) 2/7

(b) 5/7

(d) 1/7

Ans : (d) 1/7

4. Two dice are rolled simultaneously. The Probability that they show same faces is

(a) 2/3

(b) 5/6

(d) 5/6

Ans : (d) 1/6

5. In a family of 3 children, the probability of having at least one girl is

(a) 7/8

(b) 1/8

(d) 3/4

Ans : (a) 7/8

6. The probability of getting a head when a coin is tossed once is

(a) 0

(b) 1/2

(d) 1

Ans : (b) 1/2

7. Ramesh takes out all the hearts from a deck of 52 cards. The probability of picking a diamong is

(a) 1/13

(b) 1/39

(d) 1/52

Ans : (c) 1/3

8. The probability of an impossible event is

(a) 0

(b) 1/2

(d) 1

Ans : (a) 0

9. Under the usual notations in probability, P(E) + P(E) =

(a) 0

(b) 1/2

(d) None

Ans : (c) 1

10. Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is 8 ?

(a) 31/36

(b) 5/36

(d) 1

Ans : (b) 5/36

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

(a) 5/17

(b) 8/17

(d) 8/9

Ans : (b)8/17

12. From the letters of the word ’MOBILE’, a letter is selected. The probability that the letter is a vowel is

(a) 1/3

(b) 3/7

(d) 1/2

Ans : (a) 1/3

13. A month is selected at random in year. The probability that it is March or October is

(a) 1/12

(b) 1/6

(d) None

Ans : (b) 1/6

14. From a well shuffled cards, the probability of drawing a red-colored ace is

(a) 1/4

(b) 1/13

(d) 1/2

Ans : (c) 1/26

15. Which of the following does not represent probability of an event ?

(a) 0

(b) 1

(d) 0.99999

Ans : (c) 1.0001

16. ‘If a dice is thrown twice, then the number of sample events is

(a) 6

(b) 12

(d) 36

Ans : (d) 36

17. The total no. of possible outcomes when a coin is tossed

(a) 1

(b) 2

(d) Any real number

Ans : (b) 2

18. The probability of an event is 50% its value is

(a) 50

(b) 1/2

(d) Both a and b

Ans : (d) Both a and b

19. The probability of guessing the correct answer to a certain test question is x/12. If the probability

of not guessing the correct answer to this question is 2/3 then ’x’ = ?

(a) 2

(b) 3

(d) 6

Ans : (c) 4

20. A die is thrown once. The probability of getting not a prime number is

(a) 1/2

(b) 1/6

(d) None of these

Ans : (a) 1/2