How can I calculate the chance of two events both happening?
Quick answer
To find the chance that both A and B happen, use P(A and B) = P(A) × P(B given A). If A does not change the chance of B, the events are independent, so this becomes P(A) × P(B). If A changes B’s chance, use the updated conditional probability—not B’s original probability. State your assumptions and express the result as a fraction, decimal, or percentage.
The question, the details, the next step.
How do I work out the chance that two events both happen? I’m not sure whether I should multiply their probabilities.
Multiplication is the right approach, but the key is choosing the correct probability for the second event. In general, the chance that both A and B happen is P(A and B) = P(A) × P(B given A). The phrase “given A” means you calculate B’s chance in the situation where A has already happened. [S1]
If A does not affect the chance of B, the events are independent, and the rule simplifies to P(A and B) = P(A) × P(B). Do not assume that just because you know each event’s separate probability, you can multiply those two numbers. First check whether the first event changes the conditions for the second. [S1]
What does that check look like in a concrete example?
Suppose a bag contains 4 blue marbles and 3 white marbles, and you want the chance of drawing a blue marble twice. Assume each draw is equally likely to be any marble available.
If you draw without replacement, the first blue draw leaves 3 blue marbles among 6 total. The calculation is:
P(two blues) = P(first blue) × P(second blue given first blue) = 4/7 × 3/6 = 12/42 = 2/7.
The second chance is 3/6, not 4/7, because the first blue marble was removed. In this stated setup, removing a marble changes the mix, so the draws are dependent. [S2]
How would the answer change if I put the marble back?
If you replace the first marble and mix the bag before drawing again, the bag returns to its original 4-blue, 3-white mix. Assuming the draws follow that setup, the second blue probability remains 4/7. The calculation is:
P(two blues) = 4/7 × 4/7 = 16/49.
That is the independent-events version: the first draw does not change the chance of blue on the second draw. The replacement comparison illustrates why the setup matters; it does not establish independence in other situations. [S2]
Can you give me a reusable way to choose the formula and explain my answer?
Use this short decision guide:
- Define the two events precisely, including the order or time period if it matters.
- Find P(A), the chance of the first event.
- Ask: after A happens, does anything about the situation change the chance of B?
- If no, and you have a reason to treat the events as independent, calculate P(A) × P(B).
- If yes, calculate P(A) × P(B given A), using the updated conditions.
- Multiply, simplify if useful, and label the result as the chance that both events happen.
Copyable explanation: “The chance that both [A] and [B] happen is [P(A)] × [P(B given A)]. Because [brief reason the first event does/does not change the second chance], I used [conditional probability / the independent-events rule]. So the chance is [result].”
For the marble example without replacement: “The chance of two blue draws is 4/7 × 3/6 = 2/7. After the first blue marble is removed, 3 of the 6 remaining marbles are blue.” The answer depends on the stated setup and probabilities; if those are unknown or different, this calculation alone cannot establish the real-world chance.
A practical checklist
State exactly what counts as each event.
Check whether the first event changes the conditions for the second.
Use P(A) × P(B) only when independence is justified.
Otherwise use P(A) × P(B given A).
Show the calculation and label what the final probability means.
A few more useful answers
Does independent mean that the two events cannot happen together?
No. Independent events can happen together; independence means one event does not change the chance of the other. The multiplication rule for independent events gives the probability that both happen. [S1]
Can I multiply the two probabilities I was given?
Only if those are the right probabilities for the setup—for example, if the events are independent. If the first event changes the second event’s chance, use the probability of the second event given that the first occurred. [S1]
What if I do not know whether the events are independent?
Do not assume they are. Look for information about how the events are related or how the process works. If you cannot determine the conditional probability or justify independence, the information given may not be enough to calculate a definite answer.
Sources
References used for this guide.
- [S1] OpenStax Introductory Statistics: Multiplication Rule and Conditional Probability openstax.org · Checked
- [S2] OpenStax Statistics: Independent and Mutually Exclusive Events openstax.org · Checked