Probability (Term 4) Chapter Notes | Mathematics for Grade 6 PDF Download

Probability

Tossing a Coin

Coin Toss Basics

• A coin has two sides: heads (shows the country’s coat of arms) and tails (shows the coin’s value).
• Each toss has two possible outcomes: heads or tails.
• Probability: The chance of getting heads or tails is equal (1/2 or 50% each).
• Example: Tossing a coin many times results in roughly equal numbers of heads and tails due to randomness.

Comparing to Real-World Scenarios
Example: Throwing a ball against a wall and catching it:

• Success (catching) depends on skill, not equal to failing (unlike a coin toss).
• Coin tosses are purely random, with no skill involved.

Recording Data
Use a table to record coin toss results:

• Columns: Heads (tallies), Tails (tallies), Total heads, Total tails.
• Example: 10 tosses per trial, repeated 5 times (50 tosses total).
• Analyze: Expect approximately 25 heads and 25 tails, but actual results may vary slightly due to chance.

Spinner Experiments

Creating a Spinner
Spinner: A square cardboard with a pencil through the center, marked with dots on each side.
Experiment sheets (A4 paper, colored at the center):

• Sheet 1: 2/4 red, 2/4 blue (equal areas).
• Sheet 2: 1/4 red, 3/4 blue (unequal areas).
• Sheet 3: 1/4 red, 1/4 blue, 1/4 green, 1/4 yellow (equal areas).

Collecting Data
Use a 10 × 10 grid (100 squares) on squared paper for each experiment (labeled Experiment 1, 2, 3).

Process:

• Spin the spinner on the sheet’s center.
• Shade a grid square in the color where the dot lands (red, blue, etc.), from left to right, row by row, until 100 squares are shaded.

Experiments:

• Experiment 1 (Sheet 1): Expect 50 red, 50 blue (2/4 = 1/2 each).
• Experiment 2 (Sheet 2): Expect 25 red, 75 blue (1/4 red, 3/4 blue).
• Experiment 3 (Sheet 3): Expect 25 each of red, blue, green, yellow (1/4 each).

Analyzing Results

• Experiment 1: 1/2 red, 1/2 blue.
• Experiment 2: 1/4 red, 3/4 blue.
• Experiment 3: 1/4 each color.

Observations:

• Compare actual fractions (e.g., red squares ÷ 100) to expected fractions.
• Check for patterns (random vs. consistent).
• Identify runs (consecutive same-color outcomes, e.g., 3 red blocks in a row).
• Compare longest runs and their frequency across classmates’ data.

Comparison to Coin Toss:

• Coin toss (heads = red, tails = blue) mimics Experiment 1 (equal chances).
• Experiments 2 and 3 have unequal probabilities, unlike a coin.

Equivalent Experiments

• Experiment 2: Similar to a die with 1 face red, 3 faces blue (1/4 vs. 3/4).
• Experiment 3: Similar to a 4-sided die or 2 coins with outcomes mapped to red, blue, green, yellow (1/4 each).

The Subtraction Game

Game Setup

• Subtract smaller number from larger (e.g., blue 2, red 5 → 5 - 2 = 3).
• Each player picks 3 numbers from possible results (0, 1, 2, 3, 4, 5).
  Example: Player 1: 0, 4, 5; Player 2: 1, 2, 3.
• If the difference matches a player’s numbers, they score 1 point.
• Play 12 rounds; highest score wins.

Outcomes: All pairs (e.g., (1, 6), (6, 1)) give differences 0 to 5.

Analyzing Outcomes

• Use a table to list differences (e.g., row 2, column 5 → 5 - 2 = 3).
• Results: 0, 1, 2, 3, 4, 5.

Frequency:

• 0: 6 ways (e.g., (1, 1), (2, 2)).
• 1: 10 ways (e.g., (2, 1), (3, 2)).
• 2: 8 ways.
• 3: 6 ways.
• 4: 4 ways.
• 5: 2 ways.

Chances:

• For 0, 4, 5: 6 + 4 + 2 = 12/36 = 1/3.

• For 1, 2, 3: 10 + 8 + 6 = 24/36 = 2/3.

Fairness: Rules are unfair; 1, 2, 3 have twice the chance (2/3 vs. 1/3).

Making the Game Fair
Adjust rules to equalize chances:
Example: Assign numbers so each set has equal probability (e.g., split outcomes evenly).

The Addition Game

Game Setup

• Addition Game: Roll two dice, add the numbers.
• Possible results: 2 to 12 (e.g., (1, 1) = 2, (6, 6) = 12).
• Outcomes: 36 combinations (6 × 6).

Frequencies (from a table):

• 2: 1 way (1, 1).
• 3: 2 ways.
• 4: 3 ways.
• 5: 4 ways.
• 6: 5 ways.
• 7: 6 ways.
• 8: 5 ways.
• 9: 4 ways.
• 10: 3 ways.
• 11: 2 ways.
• 12: 1 way.

Single Die Expectation

• Rolling one die 10 times: Expect numbers 1 to 6 roughly equally (about 1-2 times each) due to equal probability (1/6).

Visualizing Data

• Create a pictograph to show frequencies of sums (e.g., 5 has 4 ways, mark 4 ‘x’s).
• 11 possible results (2 to 12), 36 total outcomes.

Creating a Fair Game
Fair rules: Assign sums to players so each has an equal chance to score.

• Example: Split sums (e.g., Player 1: 2, 3, 4, 11, 12; Player 2: 5, 6, 7, 8, 9, 10) to balance probabilities.
• Test fairness: Play 10 times; more plays (e.g., 30) give better fairness insight.

Points to Remember

• Probability: Chance of an outcome (e.g., 1/2 for heads in a coin toss).
• Equal probability: Coin toss (1/2 heads, 1/2 tails) or spinner with equal areas.
• Unequal probability: Spinner with unequal areas (e.g., 1/4 red, 3/4 blue).
• Run: Consecutive same outcomes (e.g., 3 red blocks).
• Fair game: Equal winning chances for players (e.g., balanced number sets).
• Outcomes: All possible results (e.g., 36 for two dice).
• Frequency: How often an outcome occurs (e.g., sum 7 in 6/36 ways).

Difficult Words

• Probability: The chance an event happens (e.g., 1/2 for heads).
• Outcome: A possible result (e.g., heads, tails).
• Run: Repeated same outcomes (e.g., 3 reds in a row).
• Fair: Giving equal chances to win.
• Spinner: A tool for random outcomes (e.g., colored areas).
• Frequency: Number of times an outcome occurs.
• Pictograph: A chart using symbols to show data.

Summary