Events and Probability

ENGAGE 1

Imagine you have a box containing 10 sticks, and [b]one[/b] of them is marked as a winning stick. You are about to draw one stick at random from the box.[br][br]a. What is the probability of drawing the winning stick?

1/5 1/10 1/2 10/1

ENGAGE 2

Explain why each stick has an equal chance of being chosen.

Because the losing sticks are marked secretly. Because the winning stick is heavier than the others. Because the box always gives out the winning stick first. Because all sticks are identical in appearance and randomly drawn.

What is probability?

[b]Probability (確率)[/b][br][br][b]Definition:[/b][br][br]Probability is a numerical value that expresses how likely an event is to occur.[br][br][br]As the number of trials increases, the observed frequency of an outcome approaches its theoretical probability.

What is "trial"?

[b]Trial (試行)[/b][br][br][b]Definition:[/b][br][br]A [b]trial[/b] is a single performance or repetition of an experiment under the same conditions.[br][br]Examples include rolling a die once, drawing a stick from a box, or flipping a coin.[br][br]Each trial produces one outcome.

What is an "event"?

[b]Event (事象)[/b][br][b]Definition:[/b][br][br]An [b]event[/b] is a specific result or a set of results from a trial.[br][br]Events are expressed as [b]subsets[/b] of the sample space (the universal set of all possible outcomes).[br][br]For example, when rolling a die:[br][br][list][*]Event A: “Rolling a 1” → A = { 1 }[br][/*] [*]Event B: “Rolling an odd number” → B = {1, 3, 5}[br][br][/*][/list]Events can be:[br][list][*][b]Certain events[/b] always occur[br][/*][*][b]Impossible events[/b]: never occur (empty set ∅)[br][/*][*][b]Elementary events[/b]: contain only one outcome[br][/*][/list]

Press "Spin" to make the spinner land on a random color.[br]The Spinner will calculate the experimental probability after each trial.[br]Press "Reset" to start over.

1. What do you think the probability should be for each color, before using the spinner?[br][br]2. How does the experimental probability compare to the anticipated (theoretical) probability?

What’s the Difference Between Experimental and Theoretical Probability?

[b]Key Difference:[/b][list][br][*][b]Experimental probability[/b] = What [b]did[/b] happen[/*][*][b]Theoretical probability[/b] = What [b]should[/b] happen (in theory)[br][/*][br][/list][br]

EXPLORE

Test 1

A bag contains 4 red, 3 blue, and 3 green marbles. One marble is drawn at random.[br][b]What is the theoretical probability of drawing a blue marble?[/b]

10% 100% 42.86% 30%

Test 2

A "trial" in probability means repeating the same action under different conditions.

FALSE TRUE

Test 3

You roll a standard 6-sided die.[br][b]List the sample space[/b] and write the set representing the event “an even number is rolled.”[br][br]a. Sample Space (U) : { }[br]b. Event A (even number) : { }

Test 4

You flip a coin once. What kind of event is “getting heads”?

Compound event Certain event Impossible event Simple event

Test 5

You draw a card from a standard deck. You say: [br]"The probability of drawing a heart is 1/4\ because there are 4 suits."[br][br][b]Is this statement an example of theoretical or experimental probability? Why?[/b]