Probability of Independent Events

Two events are independent when the outcome of one has no effect on the possible outcomes or probabilities of the other. This lesson defines independence precisely, builds the multiplication rule for combining independent probabilities, and teaches how to recognize independence from the way a problem is set up.

What you will learn

• Define independence between two events
• Multiply probabilities of independent events
• Recognize independence from a problem's setup

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

What makes two events independent?

Two events are independent exactly when knowing that one of them happened gives no new information about whether the other happened. Flipping a coin and then flipping it again are independent: the coin has no memory, so the second flip's chances of heads or tails stay the same no matter what the first flip showed.

Independence often shows up when two experiments use separate objects or separate pools that never interact — a die rolled in one hand and a coin flipped in the other, two people spinning two different spinners, or a marble drawn from a bag and then replaced before a second marble is drawn from that same bag. In every case, nothing about the first outcome changes what is possible or how likely it is for the second.

The multiplication rule for independent events

When two events $A$ and $B$ are independent, the probability that both happen is the product of their individual probabilities: $P(A\text{ and }B)=P(A)\times P(B)$. This matches the multiplication principle used for counting: if $A$ has one probability among a set of equally likely outcomes and $B$ has another among a separate set, the combined outcome's chance is the product of the two fractions.

This rule extends to any number of independent events by multiplying all of their probabilities together, in the same way the multiplication principle chains multiple independent choices. Three independent coin flips landing on heads every time has probability $\dfrac{1}{2}\times\dfrac{1}{2}\times\dfrac{1}{2}=\dfrac{1}{8}$.

Recognizing independence from a problem's setup

Certain setups reliably signal independence: two separate devices (a coin and a die, two different spinners), two separate people or locations, or a draw from a bag where the item is replaced before the next draw. "With replacement" is a common phrase that restores a bag or deck to its original state, which keeps every later draw's probabilities identical to the first.

By contrast, drawing several items from one shared pool without putting anything back changes what remains available, so those draws are not independent — that situation is covered in the next lesson. Before multiplying probabilities together, always check whether the two events could possibly affect the same pool of outcomes.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Adding the probabilities of independent events instead of multiplying them.
• Assuming two events are independent just because they involve different-looking objects, without checking whether the events actually share a pool that one outcome could change.
• Forgetting that replacing a drawn item restores the original odds, which is exactly what keeps later draws independent from earlier ones.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Two events are independent when the outcome of one does not change the probabilities of the other.
• For independent events, multiply their individual probabilities: $P(A\text{ and }B)=P(A)\times P(B)$.
• Look for setup clues like separate experiments, separate pools, or draws with replacement to recognize independence.