Probability
Grade 5-6Independent Events
Independent events do not affect each other at all, and that means the probability of both happening is found by multiplying, not adding. This lesson covers identifying whether events are independent, using the AND rule to multiply probabilities, extending the rule to three or more events, and finding a missing probability.
What you need to know
- Independent events do not affect each other probability.
- For independent events, P(A and B) equals P(A) times P(B): multiply along the branches.
- Drawing with replacement keeps events independent; drawing without replacement does not.
- Multiply for "and", add for "or".
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Identifying independent events
Two events are independent if the outcome of one has no effect on the probability of the other. Repeating an experiment with the same conditions each time, such as flipping a coin twice, gives independent results, but removing an item without replacing it changes the probabilities for what happens next, making the events not independent.
Flipping a coin and then flipping it again are independent, since the first flip does not affect the second. Picking a counter from a bag without replacing it, then picking again, are not independent, since removing the first counter changes what is left for the second pick.
Using the AND rule for independent events
When two events are independent, the probability of both happening is found by multiplying their individual probabilities together.
Events A and B are independent, with P(A) = 0.4 and P(B) = 0.5. The probability of A and B is 0.4 × 0.5 = 0.2.
Extending the AND rule to three or more events
The AND rule extends naturally to three or more independent events by multiplying all of their probabilities together.
A fair coin is flipped three times. The probability of getting heads all three times is 1/2 × 1/2 × 1/2 = 1/8.
Finding a missing probability
If P(A and B) and one of the individual probabilities are known, the other can be found by rearranging the AND rule and dividing.
Events A and B are independent, with P(A and B) = 0.12 and P(A) = 0.3. Then P(B) = 0.12 ÷ 0.3 = 0.4.
Worked examples
Three exam-style questions, fully solved
A fair coin is flipped, and then flipped again. State, giving a reason, whether the two results are independent.
Easy- 1.Consider whether the result of the first flip changes the probability of the second
- 2.Recognise that the coin has no memory of the first flip
Answer: Yes, the events are independent, since the outcome of the first flip has no effect on the second
A fair coin is flipped and a fair 6-sided dice is rolled. Find the probability of getting heads and rolling a 6.
Medium- 1.Find the probability of heads: 1/2
- 2.Find the probability of rolling a 6: 1/6
- 3.Since the events are independent, multiply the probabilities: 1/2 × 1/6
Answer: 1/12
Events A, B and C are independent. P(A) = 0.5, P(B) = 0.4, P(C) = 0.2. Find P(A and B and C).
Hard- 1.Recall that for three independent events, multiply all three probabilities together
- 2.Multiply: 0.5 × 0.4 × 0.2
Answer: 0.04
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Two independent events have P(A) = 0.5 and P(B) = 0.2. Work out P(A and B).
A fair coin is flipped twice. Work out the probability of getting two heads.
A fair dice is rolled twice. Work out the probability of getting a 6 both times.
The probability that a bus is late on any day is 0.2. Work out the probability it is late on two separate days.
Two independent events have P(A) = 0.3 and P(B) = 0.6. Work out the probability that neither event happens.
A spinner has a probability of 1/5 of landing on a winning section. It is spun twice. Work out the probability of winning both times.
Avoid these
The mistakes students make most often
Assuming events are independent without checking whether one outcome affects the other, for example missing that picking without replacement changes the probabilities.
Adding the probabilities instead of multiplying them when finding P(A and B) for independent events.
Forgetting to multiply all the probabilities together when extending the AND rule to three or more independent events.
Rearranging P(A and B) = P(A) × P(B) incorrectly when solving for a missing probability, for example multiplying instead of dividing.
Confusing independent events with mutually exclusive events, and adding probabilities instead of multiplying, or the other way round.
FAQ
Questions students and parents ask
Before this topic, make sure you know
What to learn next
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