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Lesson · Grade 8 · Mathematics

Two independent events: multiply the probabilities

  • Grade 8
  • Mathematics
  • 1 min 58 s
  • British Columbia
  • BC8-MA-41

A 6 by 2 table of every die and coin outcome shows that 1 cell in 12 is a 5 and heads. Multiplying 1/6 × 1/2 gives the same 1/12, while adding gives 2/3, which would be 8 of the 12 cells. A spinner with 4 equal parts spun twice then gives 1/4 × 1/4 = 1/16, checked as a quarter of a quarter of a 4 by 4 grid, and the video ends on the rule P(A and B) = P(A) × P(B).

Concept: Probability of two independent events: multiply

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In one sentence

The question this lesson answers

Roll a fair die and flip a fair coin. What is the probability of a 5 and heads?

These are independent events: one result does not change the chances of the other.

Transcript

What the lesson says

Roll a fair die and flip a fair coin. What is the probability of a 5 and heads?

These are independent events: one result does not change the chances of the other.

An outcome is one possible result. List them all: the die across the top, the coin down the side.

Each cell is one outcome, so there are 6 times 2, which is 12 equally likely outcomes.

Only 1 of the 12 cells is a 5 and heads, so the probability, written P, is one twelfth.

Now a shortcut. On the die alone, P of 5 is one sixth.

On the coin alone, P of heads is one half.

Multiply them. Top numbers: 1 times 1 is 1. Bottom numbers: 6 times 2 is 12. One twelfth, the same as the table.

A common mistake is adding: one sixth plus one half is two thirds, or 8 of the 12 cells. But only 1 cell works.

Now a spinner with 4 equal parts, 1 of them blue, is spun twice. What is the probability of blue both times?

The two spins are independent, and each one lands on blue with probability one quarter.

So multiply: one quarter times one quarter is one sixteenth.

Check with a grid: the first spin goes across, the second spin down, and 4 times 4 is 16 outcomes.

Blue first is a quarter of the grid, and blue again is a quarter of that quarter: 1 cell out of 16.

For independent events A and B, the probability of A and B is P of A times P of B.

Curriculum

Where this fits

British Columbia, Grade 8

Mathematics, Data and probability: theoretical probability with two independent eventsBC8-MA-41

Previous: All the Outcomes of Two Spins: Tables and Tree DiagramsNext: Is the Game Fair? Checking with Theoretical Probability

The outcomes and strands follow the British Columbia curriculum.

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