Class 10 Maths Probability – Formulas & NCERT Solutions
Class 10 Maths Chapter 14 Probability: Your Friendly Guide to Nailing It
Hey, if you're tackling Class 10 Maths and Chapter 14 on Probability feels like a roll of the dice, you're in the right place. Think of this as a chat with a friend who’s got your back—clear, no-nonsense, and packed with everything you need to master probability. It’s all about figuring out how likely something is, like the chance of rain ruining your plans or pulling a red card from a deck. We’ll cover the basics, formulas, examples, 25 NCERT questions with solutions you can toggle, and 15 FAQs from simple to tricky. Fuzzy Math Academy’s LMS and 24/7 AI Math Solver are here to back you up. Let’s dive in!
Introduction: What’s Probability All About?
Probability is just a fancy way of measuring chance. Ever wondered how likely it is to get heads in a coin toss or to pick your favorite candy from a jar? That’s what Chapter 14 teaches you to calculate. In Class 10, we focus on experimental probability (based on real trials) and theoretical probability (based on logic). It’s less about guessing and more about reasoning with numbers. By the end, you’ll be solving problems like a pro, from coin flips to card games.
Basic Rules: The Groundwork You Need
Probability is built on a few simple ideas. Here’s the deal:
- Experiment: Anything you do with uncertain outcomes, like rolling a die.
- Outcome: One possible result, like getting a 4.
- Event: A specific set of outcomes, like “rolling an even number.”
- Sample Space: All possible outcomes. For a die, it’s {1, 2, 3, 4, 5, 6}.
- Probability ranges from 0 (impossible) to 1 (certain).
- Events are mutually exclusive if they can’t happen together (e.g., getting a 1 and a 2 on one roll).
Quick tip: Always list your sample space first—it’s like mapping out your game plan.
Key Formulas: Your Probability Cheat Sheet
These are the tools you’ll use. Think of them as your shortcuts to solving problems.
Theoretical Probability
P(E) = (Number of favorable outcomes) / (Total outcomes)
In math terms: P(E) = n(E)/n(S), where E is the event, S is the sample space.
Experimental Probability
P(E) = (Number of times event occurs) / (Total trials)
Example: If heads comes up 45 times in 100 tosses, P(heads) = 45/100 = 0.45.
Complementary Events
P(not E) = 1 - P(E)
If P(rain) = 0.3, then P(no rain) = 0.7.
That’s it! These cover most Class 10 problems. Now, let’s see them in action.
Some Examples: Let’s Break It Down
Here’s how to apply those formulas, step by step, like we’re working together.
Example 1: Coin Toss (Basic)
What’s the probability of getting heads in one toss?
Sample space: {H, T}. Favorable: {H}. P(H) = 1/2 = 0.5.
Example 2: Dice Roll (Theoretical)
Probability of rolling an even number on a die?
Sample space: {1, 2, 3, 4, 5, 6}. Favorable: {2, 4, 6}. P(even) = 3/6 = 0.5.
Example 3: Cards (Slightly Trickier)
A card is drawn from a deck of 52. What’s the probability it’s a king?
Total cards = 52. Kings = 4. P(king) = 4/52 = 1/13 ≈ 0.077.
Example 4: Experimental Probability
A die is rolled 60 times; 3 appears 12 times. Find P(3).
P(3) = 12/60 = 0.2.
These examples set you up for the NCERT problems. Let’s tackle those next.
25 NCERT Questions: From Easy to Challenging (With Toggle Solutions)
These are straight from the NCERT book, starting simple and ramping up. Click to reveal solutions—try them first!
Q1 (Easy): Probability of heads in a coin toss.
Sample space: {H, T}. P(H) = 1/2 = 0.5.
Q2: P(tail) in a coin toss.
Sample space: {H, T}. P(T) = 1/2 = 0.5.
Q3: P(6) on a die roll.
Sample space: {1, 2, 3, 4, 5, 6}. P(6) = 1/6 ≈ 0.167.
Q4: P(odd number) on a die.
Favorable: {1, 3, 5}. P(odd) = 3/6 = 0.5.
Q5: P(ace) from a deck of 52 cards.
Aces = 4. P(ace) = 4/52 = 1/13 ≈ 0.077.
Q6: P(red card) from a deck.
Red cards = 26 (hearts, diamonds). P(red) = 26/52 = 0.5.
Q7: P(not a king) from a deck.
P(king) = 4/52. P(not king) = 1 - 4/52 = 48/52 = 12/13 ≈ 0.923.
Q8: Bag with 3 red, 2 blue balls. P(red).
Total = 5. P(red) = 3/5 = 0.6.
Q9: P(blue) from same bag.
P(blue) = 2/5 = 0.4.
Q10: Experimental: 50 tosses, 28 heads. P(heads).
P(heads) = 28/50 = 0.56.
Q11: P(prime number) on a die.
Primes: {2, 3, 5}. P(prime) = 3/6 = 0.5.
Q12: P(face card) from a deck.
Face cards = 12 (J, Q, K). P(face) = 12/52 = 3/13 ≈ 0.231.
Q13: P(number > 4) on a die.
Favorable: {5, 6}. P(>4) = 2/6 = 1/3 ≈ 0.333.
Q14: Bag: 4 white, 5 black, 3 green. P(not green).
Total = 12. Green = 3. P(green) = 3/12. P(not green) = 1 - 3/12 = 9/12 = 0.75.
Q15: Two coins tossed. P(both heads).
Sample space: {HH, HT, TH, TT}. P(HH) = 1/4 = 0.25.
Q16: Two coins. P(at least one head).
Favorable: {HH, HT, TH}. P(at least one H) = 3/4 = 0.75.
Q17: Die and coin. P(6 and heads).
P(6) = 1/6, P(H) = 1/2. P(both) = 1/6 * 1/2 = 1/12 ≈ 0.083.
Q18: Experimental: 100 rolls, 15 times 4. P(4).
P(4) = 15/100 = 0.15.
Q19: P(heart or king) from deck.
Hearts = 13, Kings = 4, King of hearts = 1. P(heart or king) = (13+4-1)/52 = 16/52 = 4/13 ≈ 0.308.
Q20: Three coins. P(exactly two heads).
Sample space: 8 outcomes. Favorable: {HHT, HTH, THH}. P = 3/8 = 0.375.
Q21: Bag: 6 red, 4 blue, 2 yellow. P(red or blue).
Total = 12. P(red or blue) = (6+4)/12 = 10/12 = 5/6 ≈ 0.833.
Q22: P(number divisible by 3) on die.
Favorable: {3, 6}. P = 2/6 = 1/3 ≈ 0.333.
Q23: Two dice. P(sum = 7).
Total = 36. Favorable: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. P = 6/36 = 1/6 ≈ 0.167.
Q24: Experimental: 200 trials, 80 successes. P(success).
P(success) = 80/200 = 0.4.
Q25 (Higher): Deck, P(king or queen or jack).
Kings = 4, Queens = 4, Jacks = 4. P = (4+4+4)/52 = 12/52 = 3/13 ≈ 0.231.
These questions start easy and get you thinking by the end. For visual learners, imagine a sample space as a grid (like for two dice: 6x6 = 36 outcomes). If you need a diagram, we can code a basic one, but NCERT’s examples are solid.
15 FAQs: From “What’s Probability?” to “How’s It Used?”
These are common questions I hear, from basic to brainy. Click to expand.
It’s how likely something happens, like 50% chance of rain or 1/6 for rolling a 3.
Theoretical uses logic (e.g., 1/2 for heads). Experimental uses real data (e.g., 45 heads in 100 tosses).
It’s all possible outcomes—your starting point to count favorable ones.
Nope, it’s 0 to 1. Over 1 means you messed up counting outcomes.
The opposite of your event. P(not E) = 1 - P(E). Easy shortcut.
Like two coins: List all combos (HH, HT, etc.) or multiply probabilities if independent.
Real trials have randomness. More trials, closer to theoretical.
Can’t happen together, like rolling a 1 and a 2. Add their probabilities for “or.”
Independent: One doesn’t affect the other (coin flips). Dependent: One changes the other (drawing cards without replacement).
6x6 grid: (1,1), (1,2), ..., (6,6). Total = 36 outcomes.
Like heart or king, subtract the double-counted (king of hearts) to avoid overcounting.
Missing outcomes in sample space or forgetting overlaps in “or” events.
P(A given B) = P(A and B)/P(B). Class 10 doesn’t dive deep, but it’s like narrowing your sample space.
Weather forecasts, game strategies, even insurance—anywhere you predict chances.
Probability predicts likelihood; statistics (like mean) summarizes data. Mode can hint at most likely event.
That’s the rundown on FAQs—hope they clear things up! If not, our AI solver’s got your back, 24/7.
Wrapping Up: You’re Ready to Roll
Probability’s like a game of logic—once you get the rules, it’s fun to play. Practice those NCERT questions, test yourself with the FAQs, and you’ll breeze through Chapter 14. Fuzzy Math Academy’s LMS and AI Math Solver are here for extra practice or quick checks. Got questions or want to dive into tougher stuff like conditional probability? Let us know in the comments. Keep rolling those dice!
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