Probability • Topic 1 of 4

Basic Probability

The probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes, always a value between 0 and 1. Rolling a fair die, P(even) = 3/6 = 1/2; drawing from a bag of 3 red and 5 blue marbles, P(red) = 3/8. The phrase "equally likely" is doing quiet work — the formula only holds when every outcome in the sample space has the same chance, which is why a fair coin or a well-shuffled deck is specified. When outcomes are not obviously equal, count at the level where they are: for two dice, the 36 ordered pairs are equally likely even though the sums 2 through 12 are not. Probabilities are often cleaner as fractions than decimals, and the GRE frequently accepts a fraction in a Numeric Entry box. The reliable first move on any probability question is to name the sample space — how many total outcomes — before counting the favourable ones.

✅ Solved examples

1. A bag holds 4 red, 3 green and 5 yellow balls. What is the probability of drawing a green ball?
Total = 4 + 3 + 5 = 12 balls. Favourable (green) = 3. P(green) = 3/12 = 1/4.
2. A fair die is rolled once. What is the probability of a number greater than 4?
Numbers greater than 4 are 5 and 6, so 2 favourable of 6. P = 2/6 = 1/3.
3. Two fair dice are rolled. What is the probability the sum is 7?
Total equally-likely ordered pairs = 36. Sums of 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6. P = 6/36 = 1/6.
4. Quantitative Comparison. A card is drawn from a standard 52-card deck. Quantity A: P(drawing a heart). Quantity B: P(drawing a face card — J, Q, K). Which is greater?
Hearts: 13/52 = 1/4. Face cards: 12/52 = 3/13 ≈ 0.231. Since 1/4 = 0.25 > 0.231, Quantity A is greater.

✏️ Practice — try these, take hints as needed

1. A bag has 6 white and 4 black balls. Probability of drawing white?
Total = 6 + 4 = 10.
Favourable = 6.
6/10 simplified.
3/5
2. A fair die is rolled. Probability of an odd number?
Odd numbers are 1, 3, 5.
3 of 6.
Simplify.
1/2
3. A letter is chosen at random from the word STATISTICS. Probability it is a T?
Count the letters: 10 total.
Count the T's: 3.
3/10.
3/10
4. Two fair coins are tossed. Probability of exactly one head?
Sample space: HH, HT, TH, TT.
Exactly one head: HT, TH.
2 of 4.
1/2
5. Two dice are rolled. Probability the sum is 10?
Total ordered pairs = 36.
Sum 10: (4,6),(5,5),(6,4).
3/36.
1/12

📝 Topic test — 8 questions

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