Probability • Topic 1 of 4

Classical Probability

Classical probability assumes every outcome is equally likely, so P(E) = (favourable outcomes)/(total outcomes). The whole skill is accurate counting. A single die has 6 outcomes; two dice have 36 ordered pairs; a standard pack has 52 cards (26 red, 26 black, 13 of each suit, 4 of each rank, 12 face cards). When you draw several items together, count with combinations: choosing r balls from n is C(n,r). The CAT-smart move for any "at least one" question is the complement: P(at least one) = 1 − P(none), which usually has a far shorter count. Also use the addition rule P(A∪B) = P(A) + P(B) − P(A∩B) for "A or B", subtracting the overlap so it is not counted twice. Always fix the sample space first — "with replacement" or "without replacement" silently changes the denominator.

✅ Solved examples

1. A fair die is rolled once. Find the probability of getting a number greater than 4.
Favourable = {5, 6} = 2 outcomes; total = 6. P = 2/6 = 1/3.
2. Two dice are rolled. Find the probability that the sum is 7.
Pairs summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6. Total = 36. P = 6/36 = 1/6.
3. One card is drawn from a standard pack. Find the probability it is a king or a heart.
P(king) = 4/52, P(heart) = 13/52, P(king of hearts) = 1/52. P(king∪heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
4. A bag has 4 red and 6 blue balls. Two are drawn together. Find the probability that at least one is red.
P(none red) = C(6,2)/C(10,2) = 15/45 = 1/3. P(at least one red) = 1 − 1/3 = 2/3.

✏️ Practice — try these, take hints as needed

1. A die is rolled. Probability of an even number?
Even faces are 2, 4, 6.
That is 3 favourable.
Total = 6.
1/2
2. Two dice are rolled. Probability the sum is 9?
List pairs giving 9.
(3,6),(4,5),(5,4),(6,3) = 4.
Total = 36.
1/9
3. One card is drawn from a pack. Probability it is a face card?
Face cards = J, Q, K.
3 per suit × 4 suits = 12.
Out of 52.
3/13
4. A bag has 5 white and 3 black balls. One ball is drawn. Probability it is black?
Black = 3.
Total = 8.
Ratio black/total.
3/8
5. A bag has 3 red and 5 green balls. Two drawn together. Probability both are green?
Use combinations.
C(5,2)/C(8,2).
10/28.
5/14

📝 Topic test — 8 questions

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