Class 10th Probability Notes

15

Probability

Detailed Theory • Easy English • Board Exam Focus

Step-by-Step Solutions • Formula Focus • Exam Ready

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Chapter Roadmap

1. Basic Idea of Probability 2. Random Experiment & Outcomes 3. Sample Space & Events 4. Classical Probability 5. Impossible & Sure Events 6. Complementary Events 7. Coins & Dice 8. Playing Cards 9. Board-Exam Patterns 10. 10 Detailed Solved Questions 11. Board Exam Strategy 12. Quick Revision

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Chapter Goal

Probability tells us how likely an event is to happen. This chapter focuses on identifying outcomes, writing a sample space, counting favourable outcomes and applying the classical probability formula correctly in board-exam questions.

1. Basic Idea of Probability

1.1 What is Probability?

Probability is a numerical measure of the chance that an event will happen.

Simple Meaning:

If something has a high chance of happening, its probability is closer to \(1\). If it has a very small chance, its probability is closer to \(0\).

For equally likely outcomes, the probability of an event \(E\) is found by comparing favourable outcomes with all possible outcomes.

1.2 Equally Likely Outcomes

Outcomes are equally likely when each outcome has the same chance of occurring. For example, for a fair coin, Head and Tail are equally likely.

Solved Example 1 — Coin

Question: A fair coin is tossed once. Find the probability of getting a head.

1. Possible outcomes are \(H,T\).
2. Total outcomes \(=2\).
3. Favourable outcome for Head \(=1\).
4. \(P(H)=\frac{1}{2}\).
5. Answer: \(\boxed{\frac12}\).

Solved Example 2 — Die

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

1. Possible outcomes are \(1,2,3,4,5,6\).
2. Total outcomes \(=6\).
3. Only one outcome is favourable: 5.
4. \(P(5)=\frac16\).
5. Answer: \(\boxed{\frac16}\).

Solved Example 3 — Bag

Question: A bag contains 4 red and 6 blue balls. One ball is drawn at random. Find the probability of a red ball.

1. Total balls \(=4+6=10\).
2. Favourable red balls \(=4\).
3. Therefore \(P(R)=\frac4{10}\).
4. Reduce: \(\frac4{10}=\frac25\).
5. Answer: \(\boxed{\frac25}\).

2. Random Experiment and Outcomes

2.1 Random Experiment

A random experiment is an experiment in which the exact result cannot be predicted with certainty before it is performed, although all possible results can be listed.

Examples: tossing a coin, throwing a die, drawing a card and selecting a ball from a bag.

2.2 Outcome

An outcome is one possible result of a random experiment.

Experiment	Possible Outcomes
One coin toss	H, T
One die throw	1, 2, 3, 4, 5, 6
Two coin tosses	HH, HT, TH, TT

Solved Example 1 — One Toss

Question: Write all possible outcomes when a coin is tossed once.

1. A coin has two possible faces.
2. They are Head and Tail.
3. So the outcomes are \(H\) and \(T\).
4. Answer: \(\boxed{\{H,T\}}\).

Solved Example 2 — One Die

Question: Write all possible outcomes when a standard die is thrown once.

1. A standard die has six faces.
2. The numbers are 1 to 6.
3. Answer: \(\boxed{\{1,2,3,4,5,6\}}\).

Solved Example 3 — Two Tosses

Question: Write the possible outcomes when a coin is tossed twice.

1. First toss can be H or T.
2. Second toss can be H or T.
3. List every ordered possibility.
4. Answer: \(\boxed{\{HH,HT,TH,TT\}}\).

3. Sample Space and Events

3.1 Sample Space

The sample space is the set of all possible outcomes of a random experiment. It is usually denoted by \(S\).

Event: An event is a collection of one or more outcomes from the sample space.

For one die throw:

\[ S=\{1,2,3,4,5,6\} \]

The event of getting an even number is:

\[ E=\{2,4,6\} \]

Solved Example 1 — Sample Space

Question: Write the sample space for tossing a coin once.

1. Possible outcomes are Head and Tail.
2. Therefore \(S=\{H,T\}\).
3. Answer: \(\boxed{\{H,T\}}\).

Solved Example 2 — Even Event

Question: For a die, write the event of getting an even number.

1. Sample space: \(S=\{1,2,3,4,5,6\}\).
2. Even numbers are 2, 4 and 6.
3. Therefore \(E=\{2,4,6\}\).
4. Answer: \(\boxed{\{2,4,6\}}\).

Solved Example 3 — Greater Than 4

Question: For a die, write the event of getting a number greater than 4.

1. Numbers greater than 4 are 5 and 6.
2. Therefore \(E=\{5,6\}\).
3. There are 2 favourable outcomes.
4. Answer: \(\boxed{\{5,6\}}\).

4. Classical Probability

When all outcomes are equally likely, the probability of an event \(E\) is:

\[ \boxed{P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of equally likely outcomes}}} \]

If there are \(n\) total equally likely outcomes and \(m\) favourable outcomes, then:

\[ \boxed{P(E)=\frac{m}{n}} \]

Board Tip: Write the total number of outcomes and favourable outcomes separately before applying the formula. This makes your solution clear and reduces mistakes.

Solved Example 1 — Less Than 3

Question: A die is thrown once. Find the probability of getting a number less than 3.

1. Sample space \(=\{1,2,3,4,5,6\}\).
2. Numbers less than 3 are 1 and 2.
3. Favourable outcomes \(=2\).
4. Total outcomes \(=6\).
5. \(P(E)=\frac26=\frac13\).
6. Answer: \(\boxed{\frac13}\).

Solved Example 2 — Multiple of 3

Question: A die is thrown once. Find the probability of getting a multiple of 3.

1. Multiples of 3 on a die are 3 and 6.
2. Favourable outcomes \(=2\).
3. Total outcomes \(=6\).
4. \(P(E)=\frac26=\frac13\).
5. Answer: \(\boxed{\frac13}\).

Solved Example 3 — Coloured Balls

Question: A bag contains 5 white and 7 black balls. Find the probability of selecting a white ball.

1. Total balls \(=5+7=12\).
2. Favourable white balls \(=5\).
3. \(P(W)=\frac5{12}\).
4. Answer: \(\boxed{\frac5{12}}\).

5. Impossible and Sure Events

5.1 Impossible Event

An impossible event can never occur. Its probability is zero.

\[ \boxed{P(E)=0} \]

5.2 Sure Event

A sure event must occur. Its probability is one.

\[ \boxed{P(E)=1} \]

5.3 Range of Probability

\[ \boxed{0\le P(E)\le1} \]

Solved Example 1 — Impossible

Question: Find the probability of getting 8 when a standard die is thrown.

1. A standard die has outcomes 1 to 6.
2. 8 is not a possible outcome.
3. The event is impossible.
4. Answer: \(\boxed{0}\).

Solved Example 2 — Sure Event

Question: Find the probability of getting a number from 1 to 6 on a standard die.

1. Every die outcome is from 1 to 6.
2. Therefore the event is certain.
3. Answer: \(\boxed{1}\).

Solved Example 3 — Valid Probability

Question: Can \(\frac54\) be the probability of an event?

1. Every probability must satisfy \(0\le P(E)\le1\).
2. \(\frac54=1.25>1\).
3. Therefore it cannot be a probability.
4. Answer: \(\boxed{\text{No}}\).

6. Complementary Events

If \(E\) is an event, its complementary event means that \(E\) does not occur. It is written as \(\overline E\) or \(E'\).

\[ \boxed{P(\overline E)=1-P(E)} \]

Easy Trick: When the question contains words such as “not”, “does not”, or asks for the event not occurring, check whether the complement formula can be used directly.

Solved Example 1 — Direct Complement

Question: If \(P(E)=\frac35\), find \(P(\overline E)\).

1. Use \(P(\overline E)=1-P(E)\).
2. \(P(\overline E)=1-\frac35\).
3. \(=\frac55-\frac35=\frac25\).
4. Answer: \(\boxed{\frac25}\).

Solved Example 2 — Not a Six

Question: A die is thrown once. Find the probability of not getting 6.

1. \(P(6)=\frac16\).
2. Not getting 6 is the complementary event.
3. \(P(\text{not 6})=1-\frac16\).
4. \(=\frac56\).
5. Answer: \(\boxed{\frac56}\).

Solved Example 3 — Not Red

Question: A bag contains 4 red and 6 blue balls. Find the probability of not drawing a red ball.

1. Total balls \(=10\).
2. \(P(R)=\frac4{10}=\frac25\).
3. Therefore \(P(\text{not R})=1-\frac25\).
4. \(=\frac35\).
5. Answer: \(\boxed{\frac35}\).

7. Coins and Dice — Important Board Questions

7.1 Two Coins

When a coin is tossed twice, the sample space is:

\[ \boxed{S=\{HH,HT,TH,TT\}} \]

There are 4 equally likely outcomes. Notice that \(HT\) and \(TH\) are different ordered outcomes.

7.2 Two Dice

When two dice are thrown, the total number of ordered outcomes is:

\[ \boxed{6\times6=36} \]

Solved Example 1 — Exactly One Head

Question: Two coins are tossed. Find the probability of getting exactly one head.

1. Sample space \(=\{HH,HT,TH,TT\}\).
2. Exactly one head occurs in \(HT\) and \(TH\).
3. Favourable outcomes \(=2\).
4. Total outcomes \(=4\).
5. \(P(E)=\frac24=\frac12\).
6. Answer: \(\boxed{\frac12}\).

Solved Example 2 — At Least One Head

Question: Two coins are tossed. Find the probability of getting at least one head.

1. Sample space \(=\{HH,HT,TH,TT\}\).
2. At least one head occurs in \(HH,HT,TH\).
3. Favourable outcomes \(=3\).
4. Total outcomes \(=4\).
5. \(P(E)=\frac34\).
6. Answer: \(\boxed{\frac34}\).

Solved Example 3 — Sum of 7

Question: Two dice are thrown. Find the probability of getting a sum of 7.

1. Total ordered outcomes \(=36\).
2. Pairs giving sum 7 are \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\).
3. Favourable outcomes \(=6\).
4. \(P(E)=\frac6{36}=\frac16\).
5. Answer: \(\boxed{\frac16}\).

8. Playing Cards

A standard pack contains 52 cards. It has four suits, each containing 13 cards.

Category	Number of Cards
Total cards	52
Hearts	13
Diamonds	13
Clubs	13
Spades	13
Red cards	26
Black cards	26
Aces	4
Kings	4
Queens	4
Jacks	4
Face cards (J, Q, K)	12

Solved Example 1 — Red Card

Question: Find the probability of drawing a red card.

1. Total cards \(=52\).
2. Red cards \(=26\).
3. \(P(\text{red})=\frac{26}{52}\).
4. \(=\frac12\).
5. Answer: \(\boxed{\frac12}\).

Solved Example 2 — Ace

Question: Find the probability of drawing an ace.

1. Total cards \(=52\).
2. Aces \(=4\).
3. \(P(\text{ace})=\frac4{52}\).
4. \(=\frac1{13}\).
5. Answer: \(\boxed{\frac1{13}}\).

Solved Example 3 — Face Card

Question: Find the probability of drawing a face card.

1. Face cards are J, Q and K.
2. There are \(3\times4=12\) face cards.
3. Total cards \(=52\).
4. \(P(\text{face card})=\frac{12}{52}=\frac3{13}\).
5. Answer: \(\boxed{\frac3{13}}\).

9. Board-Exam Question Patterns

Pattern 1: A die or coin is used and a particular outcome is required.

Pattern 2: An event such as even, odd, prime, multiple or number greater than a value is required.

Pattern 3: Balls of different colours are selected from a bag.

Pattern 4: A card is selected from a standard deck.

Pattern 5: The probability of an event is given and its complement is asked.

Pattern 6: Two coins or two dice are used, so careful sample-space counting is required.

10. 10 Detailed Solved Board-Style Questions

Question 1. A die is thrown once. Find the probability of getting an even number.

1. Sample space \(S=\{1,2,3,4,5,6\}\).
2. Even outcomes are \(2,4,6\).
3. Favourable outcomes \(=3\).
4. Total outcomes \(=6\).
5. \(P(E)=\frac36=\frac12\).

Answer: \(\frac12\)

Question 2. A die is thrown once. Find the probability of getting a number greater than 2 but less than 6.

1. Possible outcomes are 1 to 6.
2. Numbers greater than 2 and less than 6 are \(3,4,5\).
3. Favourable outcomes \(=3\).
4. Total outcomes \(=6\).
5. \(P(E)=\frac36=\frac12\).

Answer: \(\frac12\)

Question 3. Two coins are tossed. Find the probability of getting at least one head.

1. Sample space \(=\{HH,HT,TH,TT\}\).
2. At least one head: \(HH,HT,TH\).
3. Favourable outcomes \(=3\).
4. Total outcomes \(=4\).
5. \(P(E)=\frac34\).

Answer: \(\frac34\)

Question 4. Two dice are thrown together. Find the probability of getting a sum of 7.

1. Total ordered outcomes \(=6\times6=36\).
2. Favourable pairs: \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\).
3. Favourable outcomes \(=6\).
4. \(P(E)=\frac6{36}\).
5. \(=\frac16\).

Answer: \(\frac16\)

Question 5. A bag contains 7 red, 5 black and 8 white balls. Find the probability of getting a white ball.

1. Total balls \(=7+5+8=20\).
2. Favourable white balls \(=8\).
3. \(P(W)=\frac8{20}\).
4. Divide by 4: \(\frac8{20}=\frac25\).

Answer: \(\frac25\)

Question 6. One card is drawn from a standard deck. Find the probability of getting a king or a queen.

1. Kings \(=4\), queens \(=4\).
2. Favourable cards \(=4+4=8\).
3. Total cards \(=52\).
4. \(P(E)=\frac8{52}\).
5. Divide by 4: \(P(E)=\frac2{13}\).

Answer: \(\frac2{13}\)

Question 7. If \(P(E)=\frac7{10}\), find the probability that the event does not occur.

1. Use the complement formula \(P(\overline E)=1-P(E)\).
2. \(P(\overline E)=1-\frac7{10}\).
3. \(=\frac{10-7}{10}=\frac3{10}\).

Answer: \(\frac3{10}\)

Question 8. A card is drawn from a standard deck. Find the probability of getting neither an ace nor a king.

1. Aces \(=4\), kings \(=4\).
2. Cards that are ace or king \(=8\).
3. Cards that are neither \(=52-8=44\).
4. \(P(E)=\frac{44}{52}\).
5. Divide by 4: \(P(E)=\frac{11}{13}\).

Answer: \(\frac{11}{13}\)

Question 9. A die is thrown once. Find the probability of getting a number which is not a multiple of 3.

1. Multiples of 3 are 3 and 6.
2. \(P(\text{multiple of 3})=\frac26=\frac13\).
3. Use the complement rule.
4. \(P(\text{not multiple of 3})=1-\frac13=\frac23\).

Answer: \(\frac23\)

Question 10. A box contains 3 green, 4 yellow and 5 blue balls. Find the probability that the selected ball is neither green nor yellow.

1. Total balls \(=3+4+5=12\).
2. Neither green nor yellow means blue.
3. Favourable blue balls \(=5\).
4. \(P(E)=\frac5{12}\).

Answer: \(\frac5{12}\)

11. Board Exam Strategy

96% Marks Strategy:

• First identify the random experiment.
• Write the complete sample space whenever it is short and useful.
• Count total outcomes carefully.
• Count only those outcomes that satisfy the event.
• Use the classical formula only when outcomes are equally likely.
• For “not” questions, check \(P(\overline E)=1-P(E)\).
• For two coins, remember \(HT\) and \(TH\) are different outcomes.
• For two dice, total ordered outcomes are \(36\).
• Reduce fractions to their simplest form.
• Always check the final result using \(0\le P(E)\le1\).

12. Quick Revision

Quick Formula Sheet

Concept	Formula / Fact
Probability	\(P(E)=\frac{\text{favourable outcomes}}{\text{total equally likely outcomes}}\)
Impossible event	\(P(E)=0\)
Sure event	\(P(E)=1\)
Range	\(0\le P(E)\le1\)
Complement	\(P(\overline E)=1-P(E)\)
Two coins	\(\{HH,HT,TH,TT\}\)
Two dice	36 ordered outcomes
Cards	52 cards in a standard pack

Important Card Facts

Card Category	Number
Total	52
Red	26
Black	26
Each suit	13
Aces	4
Kings	4
Queens	4
Jacks	4
Face cards	12

Common Mistakes to Avoid

• Do not forget to count all possible outcomes.
• Do not treat \(HT\) and \(TH\) as the same outcome for two tosses.
• Do not write a probability greater than 1.
• Do not forget favourable outcomes in a multi-step experiment.
• Do not use \(1-P(E)\) for an event unless you are actually finding its complement.
• Always simplify the final fraction.

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Class 10 Mathematics • Chapter 15: Probability

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