Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
An event is a set of outcomes of a random experiment: "getting an even number" with a die groups the outcomes 2, 4 and 6. From two events A and B, we can build new ones, and that is where it gets interesting.
The complement of A, written A bar, happens exactly when A does not happen. Their probabilities complete each other: P(not A) = 1 - P(A). It is often the smartest shortcut: to compute a complicated probability, compute the complement's instead!
The event "A and B" (the intersection) happens when A and B both occur at the same time. The event "A or B" (the union) happens when at least one of the two occurs. They are linked by a key formula: P(A or B) = P(A) + P(B) - P(A and B). Why subtract? Because when adding P(A) and P(B), the outcomes shared by both events get counted twice.
When an experiment has several stages (two coin tosses, a die then a coin), the probability tree is your best ally. Each branch carries a probability, and the rule is simple: along a path, you MULTIPLY the branch probabilities. To get heads then heads: 1/2 × 1/2 = 1/4.
Two classic mistakes to avoid: adding along a path instead of multiplying, and mixing up "and" with "or". This lesson makes you work on exactly that, so these traps never catch you again.
We toss a fair coin twice. What is the probability of getting heads then heads?
arbre : P/F puis P/F -> 4 chemins
I draw the tree: the first toss has two branches (heads, tails), and each one splits into two branches for the second toss. Four paths in total: HH, HT, TH, TT.
P(P puis P) = 1/2 × 1/2 = 1/4
The heads-then-heads path goes through two branches each carrying 1/2. Along a path, I multiply: 1/2 × 1/2 = 1/4.
Result: P(heads then heads) = 1/4 = 0.25.
We know P(A) = 0.5, P(B) = 0.3 and P(A and B) = 0.2. Compute P(A or B).
P(A ou B) = P(A) + P(B) - P(A et B)
I write the union formula. The subtraction avoids double-counting the outcomes shared by A and B.
0,5 + 0,3 - 0,2 = 0,6
I plug in the values and compute: 0.8 minus 0.2 equals 0.6.
Result: P(A or B) = 0.6.