FreeLesson 1 of 36 min read
Finding the probability that two independent events both happen
Learn how to tell when two events are independent and how to multiply their probabilities to find the chance that both happen.
Many everyday questions ask about two things happening together. Will the bus be late and will it rain? Will a coin land heads and a die show a six? Once you know one simple rule, these questions become quick and reliable to answer.
What independent means
Two events are independent if one happening does not change the probability of the other. A coin toss and a die roll are independent. The coin has no way of affecting the die. Flipping the same coin twice is also independent, because the coin has no memory.
A quick test is to ask: if I knew the first result, would my chance for the second one change?
- Coin, then die: knowing the coin result tells you nothing about the die, so they are independent.
- Two rolls of a die: the first roll does not affect the second, so they are independent.
- Two cards taken from a deck without putting the first back: the deck has changed, so they are not independent.
The multiplication rule
If A and B are independent, then P(A and B) = P(A) × P(B). In words, you multiply the two probabilities.
Why multiply? Think of it as taking a fraction of a fraction. Half of all tosses give heads. Of those heads, one sixth will also come with a six on the die. One sixth of one half is 1/12.
Example: a fair coin and a fair six-sided die. Find P(heads and six).
Step 1: P(heads) = 1/2
Step 2: P(six) = 1/6
Step 3: P(heads and six) = 1/2 × 1/6 = 1/12
Check by listing every outcome:
H1 H2 H3 H4 H5 H6
T1 T2 T3 T4 T5 T6
There are 2 × 6 = 12 equally likely outcomes. Only one of them, H6, is heads and six. So the answer is 1/12, which matches.
Working with decimals
The rule works just as well with decimals. You can also use it to find the chance that neither event happens, by first working out the chance that each one does not happen.
Example (made-up figures): P(bus late) = 0.2 and P(rain) = 0.3. Assume they are independent.
Both happen: 0.2 × 0.3 = 0.06
Neither happens:
P(bus not late) = 1 - 0.2 = 0.8
P(no rain) = 1 - 0.3 = 0.7
P(neither) = 0.8 × 0.7 = 0.56
Sense check: 0.06 is smaller than both 0.2 and 0.3, as it should be.
When the rule needs care
Suppose a bag holds 3 red and 2 blue counters, and you take two without replacing the first. P(red first) = 3/5. After that, only 2 reds remain out of 4 counters, so P(red second) = 2/4. These events are not independent. You can still multiply, but you must use the updated second probability: 3/5 × 2/4 = 6/20 = 3/10.
The rule also extends to more than two independent events. The chance of three fair coins all landing heads is 1/2 × 1/2 × 1/2 = 1/8.
What to do next time
- Ask whether one event changes the chance of the other.
- If they are independent, write each probability on its own line.
- Multiply them, and simplify fractions at the end.
- Check that your answer is smaller than each separate probability.
Key takeaways
- Two events are independent when one does not change the probability of the other.
- For independent events, P(A and B) = P(A) × P(B).
- The chance of both happening is never bigger than the chance of either one alone.
Check your understanding
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