Chapter – 14

Probability

In this post we have given the detailed notes of class 10 Maths Chapter 14 (Probability) in English. These notes are useful for the students who are going to appear in class 10 board exams.

Chapter 14: Probability

Introduction to Probability

Probability is a measure of the likelihood, or chance, of an event happening or not happening. For example, when a coin is tossed, it can either land as a head or a tail — there is a chance of each of these two outcomes occurring.

In everyday life, we often use words like “probably”, “possibly” or “there is a chance” to describe uncertain events. In mathematics, probability gives us a precise, numerical way of measuring this uncertainty.

Important Terms Related to Probability

• Experiment — An action performed to obtain some result is called an experiment. For example, tossing a coin, throwing a die, or drawing a card from a deck are all experiments.
• Outcome — A possible result of an experiment. For example, getting a head is one outcome of tossing a coin.
• Event — A collection of one or more outcomes of an experiment. For example, getting an even number when a die is thrown is an event.
• Equally Likely Outcomes — Outcomes are said to be equally likely if each of them has the same chance of occurring. For example, in tossing a fair coin, getting a head and getting a tail are equally likely.

Theoretical (Classical) Definition of Probability

The theoretical probability (also called classical probability) of an event E, written P(E), is defined as:

P(E) = Number of outcomes favourable to E / Total number of possible outcomes of the experiment

This definition assumes that the outcomes of the experiment are equally likely.

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

Solution: When a coin is tossed once, there are 2 possible outcomes: Head (H) and Tail (T). Let E be the event of getting a head.

Number of outcomes favourable to E = 1. Total number of possible outcomes = 2.

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

Similarly, if F is the event of getting a tail, P(F) = P(tail) = 1/2

Elementary Event

An event having only one outcome of the experiment is called an elementary event. In the example above, both E (getting a head) and F (getting a tail) are elementary events.

Notice that P(E) + P(F) = 1/2 + 1/2 = 1. This is true in general: the sum of the probabilities of all the elementary events of an experiment is 1.

Impossible Event and Sure (Certain) Event

Impossible Event: If the probability of an event is 0, it is called an impossible event — it is impossible for such an event to occur.

Solved Example: A wooden box contains 3 blue balls and 2 red balls. Find the probability of getting a black ball.

Solution: Let A be the event of getting a black ball. There are no black balls in the box, so favourable outcomes = 0. Total possible outcomes = 3 + 2 = 5.

P(A) = 0/5 = 0. This is an example of an impossible event.

Sure (Certain) Event: If the probability of an event is 1, it is called a sure event or certain event — it is certain that such an event will occur.

Solved Example: A die is thrown once. Find the probability of getting a number greater than 0 and less than 7.

Solution: The faces of a die are marked 1, 2, 3, 4, 5, 6 — all of these numbers are greater than 0 and less than 7. So all 6 outcomes are favourable.

P(B) = 6/6 = 1. This is an example of a sure/certain event.

Note: From these two examples, we can see that the probability of an event E always satisfies: 0 ≤ P(E) ≤ 1

Complementary Events

For an event E, the event “not E” (i.e., E does not occur) is called the complement of E, denoted Ē (or E’). The events E and Ē are called complementary events.

P(E) + P(Ē) = 1, which can also be written as: P(E) = 1 − P(Ē), or P(Ē) = 1 − P(E)

Solved Example: A card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability that the card drawn is (i) an ace, (ii) not an ace.

Solution: Total possible outcomes = 52.

(i) A deck has 4 aces. Let E be the event “the card is an ace”. Favourable outcomes = 4.

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

(ii) Let F be the event “the card is not an ace”. F is the complement of E.

P(F) = 1 − P(E) = 1 − 1/13 = 12/13

Experimental (Empirical) Probability vs Theoretical Probability

Experimental (or empirical) probability is based on the actual results of an experiment that is repeated many times, i.e., it is calculated from real data collected by performing the experiment. Theoretical (classical) probability, on the other hand, is calculated by reasoning about equally likely outcomes, without actually performing the experiment. As the number of trials in an experiment becomes very large, the experimental probability of an event tends to come close to its theoretical probability.

Solved Examples on Theoretical Probability

Example 1: 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?

Solution: Total number of balls = 3 + 5 = 8. Total possible outcomes = 8.

(i) Number of red balls = 3, so favourable outcomes = 3. P(red) = 3/8

(ii) P(not red) = 1 − P(red) = 1 − 3/8 = 5/8

Example 2: 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 is (i) red, (ii) white, (iii) not green?

Solution: Total number of marbles = 5 + 8 + 4 = 17. Total possible outcomes = 17.

(i) Favourable outcomes for red = 5. P(red) = 5/17

(ii) Favourable outcomes for white = 8. P(white) = 8/17

(iii) Favourable outcomes for green = 4, so P(green) = 4/17. Hence, P(not green) = 1 − 4/17 = 13/17

Example 3: A piggy bank contains hundred 50-paise 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-paise coin, (ii) will not be a ₹5 coin?

Solution: Total number of coins = 100 + 50 + 20 + 10 = 180. Total possible outcomes = 180.

(i) Favourable outcomes for 50-paise coin = 100. P(50-paise coin) = 100/180 = 5/9

(ii) Favourable outcomes for ₹5 coin = 10, so P(₹5 coin) = 10/180 = 1/18. Hence, P(not a ₹5 coin) = 1 − 1/18 = 17/18

Example 4: Gopi buys a fish from a shop for her aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish. What is the probability that the fish taken out is a male fish?

Solution: Total number of fish = 5 + 8 = 13. Total possible outcomes = 13.

Favourable outcomes for male fish = 5. P(male fish) = 5/13

Example 5: 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.

Solution: Total possible outcomes when a die is thrown = 6, i.e., {1, 2, 3, 4, 5, 6}

(i) Prime numbers on a die: 2, 3, 5. Favourable outcomes = 3. P(prime number) = 3/6 = 1/2

(ii) Numbers lying between 2 and 6: 3, 4, 5. Favourable outcomes = 3. P(number between 2 and 6) = 3/6 = 1/2

(iii) Odd numbers on a die: 1, 3, 5. Favourable outcomes = 3. P(odd number) = 3/6 = 1/2

Example 6: Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the dice is 8.

Solution: When two dice are thrown together, the total number of possible outcomes = 6 × 6 = 36.

The pairs whose sum is 8 are: (2,6), (3,5), (4,4), (5,3), (6,2). Favourable outcomes = 5.

P(sum = 8) = 5/36

Key Facts to Remember

• The probability of a sure (certain) event is always 1.
• The probability of an impossible event is always 0.
• The probability of an event E is a number P(E), where 0 ≤ P(E) ≤ 1.
• An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.
• For any event E, P(E) + P(Ē) = 1, where Ē denotes “not E”. E and Ē are called complementary events.

Key Points to Remember

• Probability P(E) = Number of favourable outcomes ÷ Total number of possible outcomes
• 0 ≤ P(E) ≤ 1 always
• P(sure event) = 1, and P(impossible event) = 0
• P(E) + P(Ē) = 1, i.e., P(not E) = 1 − P(E)
• Sum of probabilities of all elementary events of an experiment = 1
• Theoretical (classical) probability assumes all outcomes are equally likely; experimental probability is based on repeated trials of the actual experiment
• For two dice thrown together, the total number of possible outcomes is 6 × 6 = 36

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