You may multiply P(A) and P(B) to get P(A and B) only when A and B are independent: knowing one happened does not change the chance of the other. If the events are linked, the product gives the wrong answer and the second probability must be adjusted.
This lesson sits between using a tree diagram, where independence was assumed, and removing items without replacement, where it fails.
What does independent really mean?
Independent means the probability of B is the same whether or not A happened. Tossing a coin and then rolling a die is independent: the die cannot know the coin result.
Dependent events are linked. Drawing a card and keeping it changes what is left in the pack.
Rain today often makes rain tomorrow more likely. In both cases the second probability depends on the first outcome.
How do you test independence with numbers?
Use the rule: A and B are independent exactly when P(A and B) = P(A) × P(B).
- Find P(A) and P(B).
- Find P(A and B) from the data.
- Multiply P(A) × P(B).
- Compare. Equal means independent. Not equal means dependent.
Worked example
Of 40 students in a class, 16 play football (F), 25 wear glasses (G), and 10 do both. One student is chosen at random. Are F and G independent?
Step 1. P(F) = 16/40 = 2/5 = 0.4.
Step 2. P(G) = 25/40 = 5/8 = 0.625.
Step 3. P(F and G) = 10/40 = 0.25.
Step 4, compare. P(F) × P(G) = 0.4 × 0.625 = 0.25.
Step 5, conclude. Since 0.4 × 0.625 = 0.25 = P(F and G), the events are independent. Playing football does not change the chance of wearing glasses in this class.
Check. Among the 16 football players, 10 wear glasses, which is 10/16 = 0.625. That equals P(G), which confirms that knowing a student plays football leaves the glasses probability unchanged.
The mistake to watch for
Multiplying when the events are dependent.
Mistaken working: P(rain on Monday) = 0.4 and P(rain on Tuesday) = 0.4, so P(rain both days) = 0.4 × 0.4 = 0.16.
Weather on consecutive days is linked, so this assumed independence without checking.
Correction. Suppose the forecast also says that if it rains on Monday, P(rain on Tuesday) = 0.7. Then P(rain both days) = 0.4 × 0.7 = 0.28. The second factor must be the probability of Tuesday rain given Monday rain, not the overall figure.
In an exam, state why you are multiplying: “the events are independent” (given in the question) or show the test. The word “independent” earns the method mark only when the question supports it.
Check yourself
Try these, then open each answer.
1. P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Are A and B independent?
Show answer
P(A) × P(B) = 0.5 × 0.4 = 0.2, which equals P(A and B). So yes, independent.
2. P(A) = 0.6, P(B) = 0.5 and P(A and B) = 0.4. Are A and B independent?
Show answer
P(A) × P(B) = 0.6 × 0.5 = 0.3, but P(A and B) = 0.4. They differ, so the events are not independent.
3. A and B are independent with P(A) = 0.3 and P(B) = 0.5. Find P(A and B) and P(A or B).
Show answer
P(A and B) = 0.3 × 0.5 = 0.15. P(A or B) = P(A) + P(B) − P(A and B) = 0.3 + 0.5 − 0.15 = 0.65. The subtraction is needed because A and B can happen together.
Where this leads next
Independence is what lets branch probabilities stay the same from one stage to the next. When it fails, the numbers must change, which is the subject of updating probabilities when an item is not replaced. The non-calculator working trainer checks the fraction arithmetic, and the percentage-base explorer shows why a percentage of a changed base gives a different answer.
Students who multiply correctly but leave out the reasoning often lose the explanation marks. Online one-to-one Mathematics tuition can rehearse exactly that kind of written justification.