krit.club logo

Data Handling - Chance and Probability

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A Random Experiment is an action where the result cannot be predicted with total certainty before it happens. For example, when you flip a coin, you know the result will be either Heads or Tails, but you cannot be sure which one will appear until it lands.

•

Outcomes represent the possible results of an experiment. If you roll a standard six-sided die, the outcomes are the numbers 1,2,3,4,5,1, 2, 3, 4, 5, and 66. Visually, imagine a cube with dots representing these numbers on each of its six faces.

•

An Event is a collection of one or more outcomes of an experiment. For instance, in the experiment of rolling a die, 'getting an even number' is an event consisting of outcomes 2,4,2, 4, and 66.

•

Equally Likely Outcomes occur when each outcome of an experiment has the same chance of happening. Imagine a circular spinner divided into four equal-sized sectors of different colors; because the areas are identical, the pointer is equally likely to stop on any of the colors.

•

Probability is a numerical measure of the likelihood that an event will occur, ranging from 00 to 11. On a horizontal scale, 00 represents an impossible event (like drawing a blue marble from a bag containing only red marbles), 0.50.5 represents an even chance, and 11 represents a certain event.

•

The Sample Space is the set of all possible outcomes of a random experiment. For example, if you toss two coins simultaneously, the sample space can be visualized as a grid of four pairs: (H,H),(H,T),(T,H),(H, H), (H, T), (T, H), and (T,T)(T, T).

📐Formulae

P(E)=Number of outcomes favorable to event ETotal number of possible outcomesP(E) = \frac{\text{Number of outcomes favorable to event } E}{\text{Total number of possible outcomes}}

0≤P(E)≤10 \leq P(E) \leq 1

P(Event)+P(Not Event)=1P(\text{Event}) + P(\text{Not Event}) = 1

💡Examples

Problem 1:

A bag contains 33 red marbles and 55 blue marbles. If one marble is drawn at random, what is the probability that the marble drawn is red?

Solution:

Step 1: Identify the total number of outcomes. Total marbles = 3 (red)+5 (blue)=83 \text{ (red)} + 5 \text{ (blue)} = 8. Step 2: Identify the number of favorable outcomes for the event 'drawing a red marble'. Favorable outcomes = 33. Step 3: Apply the probability formula: P(Red)=Number of red marblesTotal number of marbles=38P(\text{Red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{3}{8}

Explanation:

To find the probability, we divide the count of the specific items we are looking for (red marbles) by the total capacity of the set.

Problem 2:

When a single fair die is rolled, what is the probability of getting a prime number?

Solution:

Step 1: List the total outcomes of rolling a die: {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}. Total count = 66. Step 2: Identify the prime numbers in the outcomes: 2,3,2, 3, and 55. (Note: 11 is neither prime nor composite). Count of favorable outcomes = 33. Step 3: Calculate the probability: P(Prime)=36P(\text{Prime}) = \frac{3}{6} Step 4: Simplify the fraction: P(Prime)=12P(\text{Prime}) = \frac{1}{2}

Explanation:

This problem requires identifying the specific subset of numbers (primes) from the standard set of die faces and calculating their ratio to the total.