Maths

a = sin⁻¹ (sin5) = 5 − 2π

and b = cos⁻¹ (cos5) = 2π − 5

∴    a² + b² = (5 − 2π)² + (2π − 5)²

= 8π² − 40π + 50

The examples of the probability include - flipping a coin, rolling a die, drawing a card from a deck, and picking a ball from a bag.

There are various applications of the probability in the real life. It is used to take informed decisions and risks in the following areas - insurance, weather forecasting, finance, business, sports, computer Science and gaming.

There are two basic rules for finding the Probability of two events A and B - Multiplication Rule and Addition Rule. 

There are four types of probability - Classical, Empirical, Subjective, and Axiomatic Probability. The classical probability is based on the logical and known outcomes, Empirical are based on data from actual experiments, and subjective is based on intuition, personal opinion, or experience.

Probability in Maths is used to calculate how an event is likely to occur. It is measured by dividing the favorable outcomes number by the number of possible outcomes. It is expressed from 0% to 100% or by either 0 (impossible) or 1 (certain).

The common mistakes can be - using the formula incorrectly, before applying the formula students not simplify the expression, in Binomial Expansion, sometimes they forget to include the fractional and negative exponents, and using incorrect values for a, b, and n.

The Binomial model is known as the Lattice Model or Binomial Option Pricing Model.

There are a total of 36 questions comprising 14 straightforward formula-based sums, 16 challenging problems, and 6 intermediate.

Mathematically, binomial theorem formula is: (a + b)n = nC0 an + nC1 an - 1b + nC2 a n – 2b 2 + . + nCn – 1a.bn - 1 + nCn bn. Here a and b are real numbers and n is a positive integer.