Probability and Data Representation Concepts

Definition of Probability: The measure of the likelihood that an event will occur. Expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Sample Space:
- Definition: The set of all possible outcomes of a probabilistic experiment.
- Example: For a six-sided die, the sample space S = {1, 2, 3, 4, 5, 6}.
- Written with words: e.g., "2 red, 1 blue" translates to outcomes = {red, red, blue}.

Theoretical Probability Formula:
- P(event) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
Example:
- If there are 3 cards (2 red, 1 blue) and a friend picks a card, replaces it, and picks again, the probability of picking a red card on the first pick is:
- Favorable Outcomes = 2 (picking red)
- Total Outcomes = 3
- Thus, the probability is P(red) = \frac{2}{3} .

Numerical Representations:
- Line Plot: Visual representation using dots to show frequency of data points.
- Stem-and-Leaf Plot: Data is split into a stem (the first part) and a leaf (the last digit).
- Box Plot: Displays the distribution of data based on five summary statistics: minimum, first quartile, median, third quartile, and maximum.
- Histogram: A type of bar graph representing the frequency distribution of numerical data.
- Circle Graph (Pie Chart): Displays proportions of a whole.

Cross Multiplication Method:
- To find a percentage of a given number, use the formula:
- \text{percent given} \times X = 100\% \times \text{number given}
- Rearranging gives us X = \frac{\text{percent given} \times \text{number given}}{100\%} .

Finding Degrees for Circle Graphs:
- Given values: 6th = 96, 7th = 60, 8th = 84
- Total = 96 + 60 + 84 = 240
- For each grade:
- 6th: \frac{96}{240} \times 360 = 144°
- 7th: \frac{60}{240} \times 360 = 90°
- 8th: \frac{84}{240} \times 360 = 126°

Types: Circle graph (pie chart) or bar graph are effective for displaying categorical data, illustrating the frequency of occurrence or distribution across categories.