CBSE · Class 10 · Mathematics · Chapter 14

Probability — Formula Sheet

Board Formulas

10 formulas

1.Theoretical Probability of an Event★
$P (E) = \frac{number of outcomes favourable to E}{number of all possible outcomes of the experiment}$
$E$ : The event whose probability is required · $P (E)$ : Probability of E (a number from 0 to 1, no unit)
Valid only when all outcomes are equally likely: a fair coin, a fair die, a well-shuffled pack, a ball drawn at random.
2.Range of Probability
$0 \leq P (E) \leq 1$
A negative probability or one greater than 1 means a counting error. Recheck the total and the favourable cases.
3.Sure Event and Impossible Event
$P (sure event) = 1, P (impossible event) = 0$
Getting a number less than 7 on a die is sure (probability 1). Getting 8 on a single die is impossible (probability 0).
4.Complementary Events★
$P (E) + P (E) = 1 \Rightarrow P (not E) = 1 - P (E)$
$E$ : Complement of E, the event 'not E'
Ē (read 'not E') happens exactly when E does not. Use this when 'not E' is easier to count, or when P(E) is given and P(not E) is asked.
5.Sum over Elementary Events
$P (E_{1}) + P (E_{2}) + \dots + P (E_{n}) = 1$
An elementary event has only one outcome. The probabilities of all the elementary events of an experiment add up to 1. Use it to find one missing probability.
6.Tossing Coins
$outcomes = 2^{k} for k coins : 2, 4, 8$
Two coins: HH, HT, TH, TT. Three coins: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. 'One head and one tail' has 2 outcomes (HT, TH), not 1.
7.Throwing Dice
$one die: 6 outcomes, two dice: 6 \times 6 = 36 outcomes$
Outcomes for two dice are ordered pairs (a, b). There are 6 doublets: (1, 1), (2, 2), …, (6, 6).
8.Sum of the Numbers on Two Dice
$ways to get sum s = 6 - ∣ s - 7∣, s = 2, 3, \dots, 12$
Sum 7 has the most ways (6); sums 2 and 12 have 1 way each. Check: 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36.
9.A Pack of 52 Playing Cards★
$52 = 4 suits \times 13, face cards = 12, red = black = 26$
Spades ♠ and clubs ♣ are black; hearts ♥ and diamonds ♦ are red. Each suit has A, 2 to 10, J, Q, K. Face cards are J, Q and K only; aces are not face cards.
10.Counting Multiples from 1 to N
$multiples of k in 1, 2, \dots, N = ⌊ \frac{N}{k} ⌋$
⌊N/k⌋ is the whole-number part of N/k. Multiples of 7 from 1 to 100: ⌊100/7⌋ = 14. Useful for cards or discs numbered 1 to N.