CBSE 10
Mastering Statistics & Probability: CBSE Class 10 Math Guide
May 18, 2024
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Probability and statistics are essential branches of mathematics that help you understand uncertainty, make informed decisions, and draw meaningful conclusions from data. In this article, the fundamental concepts of probability and statistics will be explored, key terms used in these fields, important formulas, and provide solved examples to enhance your understanding. Whether you're a student or someone seeking a practical understanding of probability and statistics, this article will serve as a comprehensive guide for you.
Probability is the measure of the likelihood of an event occurring. It helps in quantifying uncertainty and make predictions based on available information. Here are some key concepts related to probability:
A random experiment is an action or process that results in an uncertain outcome. For example, tossing a coin, rolling a dice, or drawing a card from a deck.
The sample space is the set of all possible outcomes of a random experiment. For a coin toss, the sample space would be {heads, tails}.
Random variables represent numerical outcomes associated with a random experiment. For instance, the number obtained on a rolled dice or the time it takes to complete a task.
Two events are independent if the outcome of one event does not affect the outcome of the other. For example, tossing a coin and rolling a dice are independent events.
Statistics involves collecting, organizing, analyzing, interpreting, and presenting data. It helps in drawing conclusions and make informed decisions based on the information at hand. Here are some key terms related to statistics:
The mean, also known as the average, is the sum of all values in a dataset divided by the total number of values. It provides a measure of central tendency.
The expected value is the weighted average of all possible outcomes of a random experiment. It represents the long-term average that is expected to occur.
Variance measures the spread or dispersion of data points around the mean. It quantifies the variability within a dataset.
To calculate probabilities and statistical measures, various formulas are used such as:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Mean = (Sum of all values) / (Total number of values)
Variance = [(Sum of squared differences between each value and the mean) / (Total number of values)]
In a survey conducted among a group of students, their ages were recorded as follows: 18, 20, 22, 25, 21, 22, 19, 23, 20, 22. Calculate the mean and mode of the recorded ages.
✅ Solution:
To find the mean, add up all the ages and divide by the total number of ages:Mean = (18 + 20 + 22 + 25 + 21 + 22 + 19 + 23 + 20 + 22) / 10 = 212 / 10 = 21.2.
To find the mode, determine the value(s) that appear most frequently. In this case, the age 22 appears three times, which is more than any other age. Therefore, the mode is 22.
A bag contains 7 red, 5 blue, and 8 green marbles. If Shreya picks 2 marbles randomly from the bag without replacement and then picks one more marble, what is the probability that she selects 2 blue marbles and 1 red marble?
✅ Solution:
Total number of marbles = 7 (red) + 5 (blue) + 8 (green) = 20 marbles.
The probability of selecting 2 blue marbles and 1 red marble can be calculated as follows:Probability = (Number of ways to choose 2 blue marbles) * (Number of ways to choose 1 red marble) / (Total number of ways to choose 3 marbles without replacement).
Number of ways to choose 2 blue marbles = C(5, 2) = 5! / (2! * (5-2)!) = 10Number of ways to choose 1 red marble = C(7, 1) = 7! / (1! * (7-1)!) = 7Total number of ways to choose 3 marbles without replacement = C(20, 3) = 20! / (3! * (20-3)!) = 1140
Probability = (10 * 7) / 1140 = 70 / 1140 = 7 / 114.
Therefore, the probability that Shreya selects 2 blue marbles and 1 red marble is 7/114.
In a box, there are 4 white, 3 black, and 5 blue pens. If Ravi picks a pen at random, what is the probability that it is not black?
✅ Solution:
Total number of pens = 4 (white) + 3 (black) + 5 (blue) = 12 pens.
The probability of selecting a pen that is not black can be calculated as follows:Probability = Number of pens that are not black / Total number of pens.
Number of pens that are not black = 4 (white) + 5 (blue) = 9 pens.
Probability = 9 / 12 = 3 / 4.
Therefore, the probability that Ravi picks a pen that is not black is 3/4.
Calculate the mean of the following set of values: 12, 18, 15, 14, 20, 17, 11, 19, 16, 13.
✅ Solution:
To find the mean, add up all the values and divide by the total number of values:Mean = (12 + 18 + 15 + 14 + 20 + 17 + 11 + 19 + 16 + 13) / 10 = 155 / 10 = 15.5.
Therefore, the mean of the given set of values is 15.5.
A group of 25 students participated in a spelling competition, and their scores out of 20 were recorded as follows: 14, 17, 18, 16, 15, 17, 19, 12, 16, 14, 15, 16, 18, 19, 20, 16, 15, 13, 17, 16, 16, 14, 19, 20, 18. Determine the median and mode of the recorded scores.
✅ Solution:
To find the median, arrange the scores in ascending order and identify the middle value(s). In this case, the middle value(s) is between the 12th and 13th scores, which are both 16. Therefore, the median is 16.
To find the mode, determine the value(s) that appear most frequently. In this case, the score 16 appears the most, occurring 5 times. Therefore, the mode is 16.
Therefore, the median of the recorded scores is 16, and the mode is also 16.
In a box, there are 50 pencils, 30 pens, and 20 erasers. Meera randomly selects an item from the box without looking. What is the probability that she selects a ruler?
✅ Solution:
Let's assume that the box contains a total of 100 items.
Number of rulers = 0 [since the box does not contain any rulers].
Hence, the probability that Meera selects a ruler is:
P(Selecting a ruler) = Number of rulers / Total number of items
= 0/100 = 0.
Therefore, the probability of Meera selecting a ruler is 0, indicating an impossible event.
A board game has a spinner with numbered sections: 1, 2, 3, 4, 5, 6, 7, and 8. The spinner is equally likely to stop on any of these numbers. What is the probability of landing on: (i) 8, (ii) a number greater than 2, (iii) an even number?
✅ Solution:
Sample Space = {1, 2, 3, 4, 5, 6, 7, 8}.
Total number of outcomes = 8.
(i) Probability of landing on 8:
Number of ways to get 8 = 1,
P(Landing on 8) = 1/8.
(ii) Probability of landing on a number greater than 2:
Numbers greater than 2 = {3, 4, 5, 6, 7, 8},
Number of numbers greater than 2 = 6,
P(Landing on a number greater than 2) = 6/8 = 3/4.
(iii) Probability of landing on an even number:
Even numbers = {2, 4, 6, 8},
Number of even numbers = 4,
P(Landing on an even number) = 4/8 = 1/2.
Probability is the measure of the likelihood of an event occurring. It helps in quantifying uncertainty and make predictions based on available information.
Statistics involves collecting, organizing, analyzing, interpreting, and presenting data. It helps in drawing conclusions and make informed decisions based on the information at hand.
A random experiment is an action or process that results in an uncertain outcome. A random variable represents numerical outcomes associated with a random experiment.
The mean, also known as the average, is the sum of all values in a dataset divided by the total number of values. It provides a measure of central tendency.
To calculate the probability of an event, divide the number of favorable outcomes by the total number of possible outcomes. The formula is:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes).
Probability and statistics play a vital role in our understanding of uncertainty and data analysis. By grasping these concepts, terms, and formulas, you can make more informed decisions and draw meaningful conclusions from data. If you want to further enhance your knowledge and practice probability and statistics, consider subscribing to Aha, a learning website that provides quality sample questions and utilizes AI to analyze your weak points. Start your journey towards becoming a proficient problem solver in probability and statistics today!
