Complete theory, experimental probability, theoretical probability, classical definition, complementary events, impossible and sure events, playing cards, dice, coins problems, and exercise solutions for CBSE Class 10 Board Exam 2025-26.
Probability is a branch of mathematics that deals with the likelihood or chance of an event occurring. It is used extensively in weather forecasting, games, insurance, stock markets, and many real-life situations.
An experiment whose outcome cannot be predicted with certainty before it is performed. Example: Tossing a coin, rolling a die.
The set of all possible outcomes of a random experiment. Example: For a coin, S = {H, T}.
A subset of the sample space. Example: Getting a head when tossing a coin.
The number of outcomes that satisfy the event.
An event that cannot occur. Probability = 0. Example: Getting a 7 on a standard die.
An event that always occurs. Probability = 1. Example: Getting a number ≤ 6 on a die.
where n(E) = number of elements in event E, n(S) = number of elements in sample space S.
Sample Space: S = {H, T}
n(S) = 2
P(H) = 1/2, P(T) = 1/2
S = {HH, HT, TH, TT}
n(S) = 4
P(2 heads) = 1/4
P(1 head) = 2/4 = 1/2
P(0 head) = 1/4
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
n(S) = 8
P(3 heads) = 1/8
P(at least 2 heads) = 4/8 = 1/2
P(at most 1 head) = 4/8 = 1/2
Three coins are tossed simultaneously. Find the probability of getting exactly two heads.
Solution: Total outcomes = 8. Favorable outcomes: HHT, HTH, THH = 3. P = 3/8.
S = {1, 2, 3, 4, 5, 6}
n(S) = 6
P(getting 4) = 1/6
P(even number) = 3/6 = 1/2
P(prime number) = 3/6 = 1/2
Total outcomes = 36
Sum = 7 → 6 outcomes: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → P=6/36=1/6
Doublets → 6 outcomes: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) → P=6/36=1/6
Two dice are rolled. Find the probability that the sum of numbers is a prime number.
Solution: Prime sums possible: 2,3,5,7,11. Count outcomes: sum2→1, sum3→2, sum5→4, sum7→6, sum11→2. Total favorable = 15. P = 15/36 = 5/12.
A standard deck of 52 playing cards has:
One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting:
(i) a king → 4/52 = 1/13
(ii) a red card → 26/52 = 1/2
(iii) a face card → 12/52 = 3/13
(iv) a spade → 13/52 = 1/4
If the probability of an event occurring is P(E), then the probability of that event NOT occurring is 1 - P(E).
If the probability of raining tomorrow is 0.35, what is the probability of no rain?
Solution: P(no rain) = 1 - 0.35 = 0.65
Question: A coin is tossed 1000 times with the following frequencies: Head = 455, Tail = 545. Find the probability of getting a head.
Solution: P(H) = 455/1000 = 0.455 (Experimental Probability)
Question: A die is thrown once. Find the probability of getting (i) an even number, (ii) a number greater than 4.
Solution: S = {1,2,3,4,5,6}.
Even numbers: 2,4,6 → P = 3/6 = 1/2.
Number > 4: 5,6 → P = 2/6 = 1/3.
Question: One card is drawn from a well-shuffled deck of 52 cards. Find the probability of drawing a face card or a red card.
Solution: Face cards = 12, Red cards = 26. Red face cards = 6 (2 per red suit).
P(Face or Red) = (12 + 26 - 6)/52 = 32/52 = 8/13.
Question: Two dice are thrown together. Find the probability that the product of the numbers is 6.
Solution: Total outcomes = 36. Favorable: (1,6),(2,3),(3,2),(6,1) = 4. P = 4/36 = 1/9.
Question: The probability of winning a game is 0.25. What is the probability of losing the game?
Solution: P(losing) = 1 - 0.25 = 0.75.
| Situation | Sample Space (n(S)) | Example |
|---|---|---|
| 1 Coin | 2 | P(H) = 1/2 |
| 2 Coins | 4 | P(1 Head) = 1/2 |
| 3 Coins | 8 | P(All Heads) = 1/8 |
| 1 Die | 6 | P(4) = 1/6 |
| 2 Dice | 36 | P(Sum 7) = 1/6 |
| 52 Cards | 52 | P(King) = 1/13 |