Probability
REET Maths questions on Probability are built on the classical definition - P(E) = favourable outcomes / total outcomes - and simple experiments like dice and coins. Here you'll find the rules of probability, complementary events, and worked examples with solutions.
- 3 questions in REET 2024
- Class 9-10 level
- 8 min read
How REET asks this topic
All three Probability questions in REET 2024 were based on the basic rules of probability - P(E) + P(not E) = 1, the probability of a sure event, and finding the probability of getting a prime number on a die. In other words, REET doesn't test deep combinatorics, but a clear understanding of definitions and the ability to count outcomes in simple experiments (dice, coins). So when solving, carefully list out the total outcomes first, then pick out the favourable ones - never guess the answer directly.
The classical definition of probability
The classical (theoretical) probability of an event E is P(E) = number of favourable outcomes / number of total possible (equally likely) outcomes. This definition only applies when all possible outcomes are equally likely - as with a fair die or a fair coin.
Basic rules of probability
- Range of the probability of any event E
- 0 <= P(E) <= 1
- Sure event
- P(E) = 1
- Impossible event
- P(E) = 0
- Complementary event rule
- P(E) + P(not E) = 1, i.e. P(not E) = 1 - P(E)
Total outcomes in dice and coin experiments
Tossing a single coin has 2 possible outcomes (heads/tails). Tossing two coins together has 4 possible outcomes (head-head, head-tail, tail-head, tail-tail). Rolling a single die has 6 possible outcomes (1 to 6). A full deck of playing cards has 52 cards.
The prime numbers on a die are only 2, 3 and 5 - 1 is neither prime nor composite, so it is never counted among the primes. This is a very common REET trap.
Experimental vs theoretical probability
Experimental probability is found by actually carrying out an experiment: number of times an event occurs / total number of trials. It can change from trial to trial. Theoretical probability is worked out purely by logical calculation, without any experiment. As the number of trials gets very large, experimental probability starts approaching theoretical probability.
Sample space and event
The set of all possible outcomes of an experiment is called the Sample Space (S). For example, rolling a die gives S = {1, 2, 3, 4, 5, 6}, and tossing two coins gives S = {HH, HT, TH, TT}. Any subset of this sample space is called an Event - for example, 'getting an even number' is the subset {2, 4, 6} of the die's sample space.
Rolling two dice together gives a total of 36 (6x6) possible outcomes, because each die's 6 outcomes combine with the other's. Example: find the probability that the sum of the numbers on both dice is 7. The pairs that add up to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - a total of 6 favourable outcomes. So probability = 6/36 = 1/6.
Similarly, a deck of 52 playing cards has 4 suits (spades, clubs, hearts, diamonds), 13 cards in each suit, and a total of 26 red cards (hearts, diamonds) and 26 black cards (spades, clubs) - this information often comes in handy in card-based probability questions.
Quick reference: common probability examples
| Event | Favourable outcomes | Probability |
|---|---|---|
| Getting an even number on a die | 2, 4, 6 (3 in total) | 1/2 |
| Getting an odd number on a die | 1, 3, 5 (3 in total) | 1/2 |
| Getting a prime number on a die | 2, 3, 5 (3 in total) | 1/2 |
| Getting heads on a coin toss | Heads (1 in total) | 1/2 |
| Drawing an Ace from 52 cards | 4 aces | 1/13 |
| Drawing a red card from 52 cards | 26 red cards | 1/2 |
All these examples assume a fair die, a fair coin, and a well-shuffled deck of cards.
How to teach this to children
Classroom activity
Have students actually toss a real coin or roll a real die repeatedly and record the results in a table in class, then work out the experimental probability and compare it with the theoretical probability (1/2 or 1/6). Combine results from the whole class to show that as the number of trials increases, the experimental value gets closer to the theoretical value.
Teaching aids
A real die, a coin, and a bag filled with coloured balls - have students pull out a ball without looking and discuss the possibilities. You can also build the sample space of the sum from rolling two dice together as a table.
Common misconception
Students often think 1 is a prime number, and some believe the probability of an event can be more than 1 if it is 'very likely' - whereas it is always bounded between 0 and 1. Some children assume all sums from rolling two dice (2 to 12) are equally likely, when in fact the number of ways to make each sum is different. Some students also think that past outcomes affect future ones (for example, that after three heads in a row, tails is now 'more likely') - whereas each toss is an independent event.
Assessment idea
Put balls of different colours in a bag and have students draw without looking to find the probability of each colour - a practical, hands-on assessment activity, which can later be compared against the actual counted ratio.
Trick to remember / exam tip
1 is neither prime nor composite - the prime numbers on a die are only 2, 3 and 5 (3 favourable outcomes in total). This mistake is caught again and again in REET, so never count 1 as prime. Also remember that every toss of a coin or roll of a die is an independent event - the previous outcome has no effect on the probability of the next one.
Questions asked in REET 2024 (with solutions)
Question 103
P(E) + P(Ē) equals :
- A0
- B2
- C1
- D3
Explanation
Ē घटना E की पूरक (complementary) घटना है, अर्थात् 'E का न होना'। किसी प्रयोग में या तो E होगी या नहीं होगी, इसलिए दोनों की प्रायिकताओं का योग निश्चित घटना की प्रायिकता 1 के बराबर होता है: P(E) + P(Ē) = 1।
Question 106
The probability of a certain event will be
- A1
- B0
- C1/2
- D1/3
Explanation
निश्चित घटना (sure/certain event) वह है जो हर बार अवश्य घटित होती है, अतः उसकी प्रायिकता 1 होती है। असंभव घटना की प्रायिकता 0 होती है। किसी भी घटना की प्रायिकता 0 और 1 के बीच होती है।
Question 112
The probability of getting a prime number on its upper face, in one toss of a dice will be
- A1/2
- B2/3
- C1/3
- D5/6
Explanation
पासे पर संभावित परिणाम 1, 2, 3, 4, 5, 6 (कुल 6) हैं। इनमें अभाज्य संख्याएँ 2, 3 और 5 (कुल 3) हैं; 1 अभाज्य नहीं है। अतः प्रायिकता = 3/6 = 1/2।
Practice Questions
Question 1
What is the probability of getting a head when a coin is tossed once?
- A1/4
- B1/3
- C1/2
- D1
Explanation
Total outcomes = 2 (heads, tails), favourable outcomes = 1 (heads). P(heads) = 1/2.
Question 2
Find the probability of getting an even number when a die is thrown.
- A1/3
- B1/2
- C2/3
- D5/6
Explanation
Even numbers are 2, 4, 6 (3 of them), total outcomes = 6. P = 3/6 = 1/2.
Question 3
A bag contains 3 red and 5 blue balls. Find the probability of drawing a red ball without looking.
- A3/8
- B5/8
- C3/5
- D1/3
Explanation
Total balls = 3 + 5 = 8, favourable (red) = 3. P(red) = 3/8.
Question 4
A card is drawn from a well-shuffled deck of 52 cards. Find the probability that it is an Ace.
- A1/13
- B1/26
- C1/52
- D4/13
Explanation
A deck has 4 aces in total. P(ace) = 4/52 = 1/13.
Question 5
If the probability of an event E is P(E) = 0.35, find P(not E).
- A0.65
- B0.35
- C1.35
- D0
Explanation
P(E) + P(not E) = 1, so P(not E) = 1 - 0.35 = 0.65.
Question 6
Two coins are tossed together. Find the probability that both show heads.
- A1/2
- B1/4
- C1/3
- D3/4
Explanation
Total outcomes = 4 (HH, HT, TH, TT), favourable outcomes = 1 (HH). P = 1/4.
Question 7
Find the probability of getting a number greater than 4 when a die is thrown.
- A1/2
- B1/6
- C1/3
- D2/3
Explanation
Numbers greater than 4 are 5, 6 (2 of them), total outcomes = 6. P = 2/6 = 1/3.
Question 8
A number is randomly chosen from 1 to 20. Find the probability that it is divisible by 3.
- A3/10
- B1/4
- C2/5
- D1/5
Explanation
Multiples of 3 from 1 to 20: 3, 6, 9, 12, 15, 18 (6 of them). P = 6/20 = 3/10.
Question 9
The probability of an event cannot be 1.2 because:
- Aप्रायिकता सदैव 0 और 1 के बीच होती है
- B1.2 एक भिन्न (fraction) है
- Cपासे पर 1.2 अंक नहीं होता
- Dयह एक सम संख्या नहीं है
Explanation
The probability of any event can only lie between 0 and 1, inclusive, since favourable outcomes can never exceed total outcomes.
Question 10
Which is the most effective way to help students understand the difference between experimental and theoretical probability?
- Aकेवल परिभाषा बोर्ड पर लिखना
- Bवास्तव में सिक्का/पासा बार-बार उछलवाकर परिणाम दर्ज कराना और सैद्धांतिक मान से तुलना कराना
- Cप्रश्नों का सीधा उत्तर बताना
- Dविषय को छोड़ देना
Explanation
Actually carrying out the experiment and recording results, then comparing them with the theoretical value, helps students understand the difference between the two concepts through direct experience.
Frequently Asked Questions
How many questions come from Probability in REET?+
REET 2024 had 3 questions from this topic, mostly based on simple dice-and-coin experiments and basic rules (like the complementary event), not deep combinatorics.
What is the classical definition of probability?+
P(E) = number of favourable outcomes / number of equally likely total outcomes - this applies only when all outcomes are equally likely, as with a fair die or coin.
Why can the probability of an event never be negative or greater than 1?+
Because the number of favourable outcomes can never be negative and can never exceed the total number of outcomes, so probability always stays between 0 and 1, inclusive.
Which numbers on a die are prime?+
On a die (1 to 6), the prime numbers are 2, 3 and 5. 1 is not considered prime, since it is neither prime nor composite - this is a common REET trap.
Why can probability be written as a fraction, decimal or percentage?+
Because probability is fundamentally a ratio, it can be correctly expressed as a fraction (like 1/2), a decimal (0.5) or a percentage (50%) - all three are equally valid. In everyday life, this idea shows up in things like weather forecasts and the chances of winning in sports.
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- 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked