Probability — Class 10 Maths Notes
Probability · Class 10 Maths · 16 topics.
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Topics covered in Probability
1.Introduction
Probability is a way to measure how likely something is to happen. Imagine you have a bag with 5 red balls and 5 blue balls. If you close your eyes and pick a ball, what are the chances you'll get a red ball? Probability helps us figure this out.
The basic formula for probability is:
Probability=Number of favorable outcomesTotal number of possible outcomesProbability=Total number of possible outcomesNumber of favorable outcomes
In our example, the favorable outcome is picking a red ball. There are 5 red balls, so the number of favorable outcomes is 5. The total number of balls is 10 (5 red + 5 blue). So, the probability of picking a red ball is:
Probability of red ball=510=0.5Probability of red ball=105=0.5
This means there's a 50% chance of picking a red ball.
Real-Life Example: Let's say you have a deck of cards (52 cards in total). You want to know the probability of picking an Ace. There are 4 Aces in the deck. So, the probability of picking an Ace is:
Probability of an Ace=452=113Probability of an Ace=524=131
This is useful in games like poker or blackjack, where knowing the odds can help you make better decisions.
Careers Using Probability:
- Statisticians use probability to analyze data and make predictions.
- Game designers use it to make games fair and exciting.
- Financial analysts use probability to assess risks and make investment decisions.
2.A bag contains a red ball, a blue ball and a yellow ball, all the balls being of the same size. Kritika takes out a ball from the bag without looking into it. What is the probability that she takes out the (i) yellow ball? (ii) red ball? (iii) blue ball?
To solve these probability problems, we use the same basic formula:
Probability=Number of favorable outcomesTotal number of possible outcomesProbability=Total number of possible outcomesNumber of favorable outcomes
In this scenario, the bag contains 3 balls: 1 red, 1 blue, and 1 yellow. So, the total number of possible outcomes is 3.
Probability of taking out the yellow ball:
- There is 1 yellow ball.
- Total balls = 3.
- Probability = 1331.
Probability of taking out the red ball:
- There is 1 red ball.
- Total balls = 3.
- Probability = 1331.
Probability of taking out the blue ball:
- There is 1 blue ball.
- Total balls = 3.
- Probability = 1331.
So, for each ball, the probability of Kritika taking it out is 1331
3.If P(E) = 0.05, what is the probability of ‘not E’?
The probability of an event "E" occurring is represented as P(E). The probability of the event "not E" happening, which means the event E does not occur, is represented as (not)P(not E).
In probability theory, the sum of the probabilities of all possible outcomes must equal 1. So, if the probability of E happening is 0.05 (or 5%), then the probability of E not happening is:
(not)=1−P(not E)=1−P(E) (not)=1−0.05P(not E)=1−0.05 (not)=0.95P(not E)=0.95
So, the probability of "not E" is 0.95, or 95%.
4.A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out (i) an orange flavoured candy? (ii) a lemon flavoured candy?
In this situation, we know the bag contains only lemon-flavored candies. Let's analyze the probabilities:
Probability of taking out an orange-flavored candy:
- Since the bag contains only lemon-flavored candies, there are no orange-flavored candies in the bag.
- This means the probability of taking out an orange-flavored candy is 0. There is no chance of this happening.
Probability of taking out a lemon-flavored candy:
- All candies in the bag are lemon-flavored.
- Therefore, the probability of taking out a lemon-flavored candy is 1 (or 100%). It is certain to happen because there are no other types of candies in the bag.
5.Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?
Tossing a coin is considered a fair way to decide which team gets the ball at the beginning of a football game because it is a random process that gives both teams an equal chance of winning. Here's why:
Equal Probability: A standard coin has two sides – heads and tails – and each has an equal probability of coming up when the coin is tossed. This means there's a 50% chance for heads and a 50% chance for tails.
Unpredictability: It is nearly impossible to predict the outcome of a coin toss. Factors like the force of the toss, the air resistance, and the landing surface all contribute to the randomness of the result.
Simplicity and Speed: Coin tossing is a quick and simple method. It doesn't require any special equipment, skills, or extensive time, making it an efficient way to make a fair decision at the start of the game.
Neutral and Unbiased: The coin itself has no preference or bias towards either team. It's an inanimate object that doesn't favor one side over the other, ensuring fairness.
These factors make coin tossing a widely accepted and fair method to decide which team starts with the ball in many sports, including football.
6.It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?
To find the probability that two students in a group of three have the same birthday, we can use the complement rule in probability. The complement rule states that the probability of an event not happening is 1 minus the probability of the event happening.
Given:
- The probability of two students not having the same birthday is 0.992.
We need to find:
- The probability that two students have the same birthday.
Using the complement rule: (same birthday)=1−(not same birthday)P(same birthday)=1−P(not same birthday) (same birthday)=1−0.992P(same birthday)=1−0.992 (same birthday)=0.008P(same birthday)=0.008
So, the probability that two out of three students have the same birthday is 0.008, or 0.8%.
7.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 will be (i) red ? (ii) white ? (iii) not green?
Let's calculate these probabilities using the formula for probability:
Probability=Number of favorable outcomesTotal number of possible outcomesProbability=Total number of possible outcomesNumber of favorable outcomes
In this case, the box contains 5 red marbles, 8 white marbles, and 4 green marbles. So, the total number of marbles is 5 (red) + 8 (white) + 4 (green) = 17 marbles.
Probability of taking out a red marble:
- Number of red marbles = 5.
- Total marbles = 17.
- Probability of red = 517175.
Probability of taking out a white marble:- Number of white marbles = 8.
- Total marbles = 17.
- Probability of white = 817178.
Probability of not taking out a green marble:- Number of non-green marbles = red + white = 5 + 8 = 13.
- Total marbles = 17.
- Probability of not green = 13171713.
So, the probabilities are:
- Red marble: 517175
- White marble: 817178
- Not green marble: 13171713
8.A piggy bank contains hundred 50p coins, fifty Rs, 1 coins, twenty Rs. 2 coins and ten Rs. 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 piggy bank contains hundred 50p coins, fifty Rs. 1 coins, twenty Rs. 2 coins and ten Rs.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 p coin ? (ii) will not be a Rs.5 coin?
To calculate the probabilities, let's first determine the total number of coins in the piggy bank and then use the probability formula:Probability=Number of favorable outcomesTotal number of possible outcomesProbability=Total number of possible outcomesNumber of favorable outcomes
Total number of coins:
- 100100 fifty-paise coins
- 5050 one-rupee coins
- 2020 two-rupee coins
- 1010 five-rupee coins
- Total = 100+50+20+10=180100+50+20+10=180 coins
Probability of getting a 50 paise coin:- Number of favorable outcomes (fifty-paise coins) = 100100
- Total number of coins = 180180
- Probability = 100180=1018180100=1810
Probability of not getting a Rs. 5 coin:- Number of coins that are not Rs. 5 = 100100 (fifty-paise) + 5050 (one-rupee) + 2020 (two-rupee) = 170170
- Total number of coins = 180180
- Probability = 170180=1718180170=1817
So, the probabilities are:
- Getting a 50 paise coin: 10181810
- Not getting a Rs. 5 coin: 17181817
9.Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish (see Fig.). What is the probability that the fish taken out is a male fish?
To find the probability that Gopi gets a male fish, we use the basic probability formula:
Probability=Number of favorable outcomesTotal number of possible outcomesProbability=Total number of possible outcomesNumber of favorable outcomes
The tank contains 5 male fish and 8 female fish, making a total of 5 + 8 = 13 fish.
Now, we calculate the probability of taking out a male fish:
- Number of favorable outcomes (male fish) = 5
- Total number of possible outcomes (total fish) = 13
Using the formula, the probability of taking out a male fish is:
(male fish)=513P(male fish)=135
So, the probability that the fish taken out is a male fish is 513135.
10.A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Fig. ), and these are equally likely outcomes. What is the probability that it will point at (i) 8 ? (ii) an odd number? (iii) a
To solve this probability problem, we can apply the basic probability formula:
Probability=Number of favorable outcomesTotal number of possible outcomesProbability=Total number of possible outcomesNumber of favorable outcomes
Given the game has numbers 1 through 8, and each outcome is equally likely, the total number of possible outcomes is 8.
Probability it will point at 8:
- Number of favorable outcomes (pointing at 8) = 1.
- Probability = 1881.
Probability it will point at an odd number (1, 3, 5, 7):
- Number of favorable outcomes (odd numbers) = 4.
- Probability = 48=1284=21.
Probability it will point at a number greater than 2 (3, 4, 5, 6, 7, 8):
- Number of favorable outcomes (numbers greater than 2) = 6.
- Probability = 68=3486=43.
Probability it will point at a number less than 9 (which includes all the given numbers):
- Number of favorable outcomes (numbers less than 9) = 8.
- Probability = 88=188=1.
So, the probabilities are:
- Pointing at 8: 1881 or 12.5%.
- Pointing at an odd number: 1221 or 50%.
- Pointing at a number greater than 2: 3443 or 75%.
- Pointing at a number less than 9: 11 or 100%.
11.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
Let's calculate the probability for each of the given scenarios. A standard die has six faces, numbered from 1 to 6.
A prime number: The prime numbers on a die are 2, 3, and 5.
- Number of favorable outcomes (prime numbers) = 3 (2, 3, 5)
- Total number of possible outcomes (faces on a die) = 6
- Probability of getting a prime number = 36=1263=21
A number lying between 2 and 6: The numbers lying between 2 and 6 are 3, 4, and 5.
- Number of favorable outcomes (numbers between 2 and 6) = 3 (3, 4, 5)
- Probability of getting a number between 2 and 6 = 36=1263=21
An odd number: The odd numbers on a die are 1, 3, and 5.
- Number of favorable outcomes (odd numbers) = 3 (1, 3, 5)
- Probability of getting an odd number = 36=1263=21
In each case, the probability is 1221, which means there's a 50% chance for each of these events to occur when a die is thrown once.
12.One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour (ii) a face card (iii) a red face card (iv) the jack of hearts (v) a spade (vi) the queen of diamonds
(i) Probability of getting a king of red color:
- Formula: Probability = (Number of favorable outcomes) / (Total number of outcomes)
- Favorable outcomes: 2 red kings
- Total outcomes: 52 cards
- Probability = 2/52 = 1/26
(ii) Probability of getting a face card:
- Favorable outcomes: 12 face cards (4 kings, 4 queens, and 4 jacks)
- Probability = 12/52 = 3/13
(iii) Probability of getting a red face card:
- Favorable outcomes: 6 red face cards (2 red kings, 2 red queens, and 2 red jacks)
- Probability = 6/52 = 3/26
(iv) Probability of getting the jack of hearts:
- Favorable outcomes: 1 jack of hearts
- Probability = 1/52
(v) Probability of getting a spade:
- Favorable outcomes: 13 spades
- Probability = 13/52 = 1/4
(vi) Probability of getting the queen of diamonds:
- Favorable outcomes: 1 queen of diamonds
- Probability = 1/52
13.Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. (i) What is the probability that the card is the queen? (ii) If the queen is drawn and put aside, what is the pro
Let's calculate the probabilities step by step.
(i) Probability that the card picked is the queen:
- There is only one queen of diamonds among the five cards.
- The total number of cards is 5 (ten, jack, queen, king, and ace of diamonds).
- So, the probability of picking the queen is 1/5.
(ii) If the queen is drawn and put aside:
(a) Probability that the second card picked is an ace:
- After removing the queen, there are now four cards left (ten, jack, king, and ace).
- There is one ace among these four remaining cards.
- So, the probability of picking an ace as the second card is 1/4.
(b) Probability that the second card picked is a queen:
- After removing the queen, there are four cards left (ten, jack, king, and ace).
- Since the queen has already been removed, there is no queen left in the remaining cards.
- So, the probability of picking a queen as the second card is 0 (impossible).
To summarize:
(i) Probability of picking the queen initially: 1/5
(ii) (a) Probability of picking an ace as the second card: 1/4
(b) Probability of picking a queen as the second card: 0
14.12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.
To determine the probability that the pen taken out is a good one, we can use the following formula:
Probability (Good Pen) = (Number of Good Pens) / (Total Number of Pens)
In this case, we have 12 defective pens mixed with 132 good ones, so:
Number of Good Pens = 132 Total Number of Pens = 12 (defective pens) + 132 (good pens) = 144
Now, we can calculate the probability:
Probability (Good Pen) = 132 / 144
To simplify this fraction, we can find the greatest common divisor (GCD) of 132 and 144, which is 12. Then, we can divide both the numerator and denominator by 12:
Probability (Good Pen) = (132 / 12) / (144 / 12) Probability (Good Pen) = 11 / 12
So, the probability that the pen taken out is a good one is 11/12.
15.A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5.
Let's calculate the probabilities for each of these scenarios:
(i) Probability of drawing a disc with a two-digit number:
- The two-digit numbers are from 10 to 90 (inclusive), which means there are 90 - 10 + 1 = 81 two-digit numbers in the box.
- The total number of discs in the box is 90.
- So, the probability of drawing a disc with a two-digit number is 81/90.
(ii) Probability of drawing a disc with a perfect square number:
- Perfect square numbers from 1 to 90 are 1, 4, 9, 16, 25, 36, 49, and 64.
- There are 8 perfect square numbers in the box.
- The total number of discs in the box is 90.
- So, the probability of drawing a disc with a perfect square number is 8/90.
(iii) Probability of drawing a disc with a number divisible by 5:
- Numbers divisible by 5 in the range from 1 to 90 are 5, 10, 15,..., 85, 90.
- There are 18 numbers divisible by 5 in the box.
- The total number of discs in the box is 90.
- So, the probability of drawing a disc with a number divisible by 5 is 18/90.
To summarize:
(i) Probability of drawing a two-digit number: 81/90
(ii) Probability of drawing a perfect square number: 8/90
(iii) Probability of drawing a number divisible by 5: 18/90
16.Quick Revision
Probability in mathematics is about measuring how likely something is to happen. Think of it as predicting whether it will rain tomorrow or if you will pick a red card from a deck of playing cards.
Key Concepts:
- Experiment: An action you do to see what happens (e.g., flipping a coin).
- Outcome: The result of an experiment (e.g., heads or tails).
- Sample Space: All possible outcomes (e.g., heads and tails).
- Event: A specific outcome or set of outcomes we're interested in (e.g., getting heads).
Important Formulas:
- Probability of an Event: =Number of favorable outcomesTotal number of outcomesP(Event)=Total number of outcomesNumber of favorable outcomes
- Example: Probability of getting heads in a coin flip is 1221 because there are 2 outcomes (heads and tails), and only 1 is favorable (heads).
How to Use the Formula:
- Identify the total number of outcomes.
- Count how many of these are favorable to your event.
- Divide the number of favorable outcomes by the total number of outcomes.
This is useful in real life to predict outcomes in situations like weather forecasting, games, and even in business decisions.