lesson
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If you flip a fair coin and get heads 5 times in a row, is tails "due" on the 6th flip?
It feels like tails should happen next, but the coin has no memory — the chance of heads is still exactly 50%.
What Makes Events Independent?
Two events are independent events when the occurrence of one event has zero effect on the probability of the other event.
If knowing that event A happened gives you no new information about the chance of event B, then A and B are completely independent.
📊Create an interactive visual comparison between Independent and Dependent events. Left card: 'Independent Events' featuring two separate spinners (Spinner 1: Coin Flip, Spinner 2: 6-Sided Die). An arrow indicates 'Spinning a 4 does not change the coin's 50% probability'. Right card: 'Dependent Events' showing a jar with 3 blue and 2 red marbles. An animated hand draws a blue marble without replacing it, showing the total dropping from 5 to 4 marbles, changing the next draw's probability from 3/5 to 2/4. Clean modern UI, slate dark text #1e2945, soft blue #3b82f6 highlights, crisp border cards.
How do we write this rule in the language of mathematics?
The Multiplication Rule for Independence
In the 1650s, mathematicians Blaise Pascal and Pierre de Fermat studied gambling games and proved that when two trials do not influence each other, their combined likelihood multiplies.
Two events A and B are independent if and only if the probability of both occurring together equals their product: P(A and B)=P(A)⋅P(B)
📊Create a visual area-model grid showing why probabilities multiply for independent events. Event A: Flipping Heads on a coin (P(A) = 1/2, split horizontally). Event B: Rolling a 6 on a fair die (P(B) = 1/6, split vertically into 6 columns). The grid displays 12 equal rectangles (2 rows x 6 columns). The single top-right box represents 'Heads AND Roll 6', highlighted in vibrant emerald green (#10b981) labeled '1 out of 12 = 1/2 × 1/6 = 1/12'. Responsive, neat mathematical layout.
What does this look like in a real problem?