Probability: classification of events, coins, dice and cards · PEA515

Probability = (number of favourable outcomes) / (total number of equally likely outcomes), a number from 0 to 1. The probability of an event not happening is 1 - P(event) (the complement rule), which is the quickest route for "at least one" questions.

Experiment	Useful facts
Two dice	36 outcomes; a sum of 7 has 6 ways, so 1/6
Standard pack	52 cards: 13 of each suit, 4 kings, 12 face cards, 26 red
n coin tosses	`2^n` outcomes; at least one head = `1 - 1/2^n`
Balls without replacement	Count selections with combinations: `C(r, 2) / C(n, 2)`
A or B	`P(A) + P(B) - P(A and B)`

Example: the probability that a card is a face card or a heart is (12 + 13 - 3) / 52 = 22/52 = 11/26, because the three face cards of hearts were counted twice. For two throws of a die, P(at least one six) = 1 - (5/6)^2 = 11/36.

Exam trap: when two events overlap, adding their probabilities counts the overlap twice, so subtract it once.

Try two questions

Question 1 · Tier 4, Analyse

A bag contains 4 red, 3 blue and 1 green balls. Two balls are drawn at random without replacement. What is the probability that both are red?

1/4
3/14
1/2
3/16

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B. Worked answer: C(4, 2) / C(8, 2) = 3/14.

Why not A: Treats the draws as with replacement.

Why not C: Gives the probability for a single draw.

Why not D: Uses n for both denominators.

Question 2 · Tier 2, Understand

One card is drawn at random from a standard pack of 52 cards. What is the probability that it is a king or a spade?

17/52
1/52
4/13
16/51

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C. 4 + 13 - 1 = 16 favourable cards out of 52: 4/13.

Why not A: Adds the two counts without removing the cards counted twice.

Why not B: Multiplies the probabilities as if the events were independent and both required.

Why not D: Divides by 51 instead of 52 cards.

Key points

Try two questions