Maths

a + d, a + 7d and a + 43d are 1^st, 2^nd, 3^rd term of G.P.

$\frac{a+7d}{a+d}=\frac{a+43d}{a+7d}$              

⇒ (a + 7d)² = (a + d) (a + 43d)

⇒ a² + 49d² + 14d = a² + 44ad + 43d³

⇒ 6d² = 30ad

⇒ d² = 5d

⇒ d = 0, 5

a = 1, d = 5

  $S_{20}=\frac{20}{2}\left[2+\left(19\right)5\right]$          

= 10 [95 + 2]

= 970

  $\frac{a+b+68+44+40+60}{6}=55$

212 + a + b = 330

⇒ a + b = 118

$\frac{{\displaystyle \sum _{}^{}{x}_i^2}}{n}-{\left(\overline{x}\right)}^2=194$          

$\frac{a^2+b^2+{\left(68\right)}^2+{\left(44\right)}^2+{\left(40\right)}^2+{\left(60\right)}^2}{6}={\left(55\right)}^2=194$

= 3219

11760 + a² + b² = 19314

⇒ a² + b² = 19314 – 11760

= 7554

(a + b)² –2ab = 7554

From here b = 41.795

a + b = 118

⇒ a + b + 2b = 118 + 83.59

= 201.59

Let probability of tail is   $\frac{1}{3}$

⇒ Probability of getting head = $\frac{2}{3}$  

∴ Probability of getting 2 heads and 1 tail

$=\left(\frac{2}{3}*\frac{2}{3}*\frac{1}{3}\right)*3$

$=\frac{4}{27}*3$

$=\frac{4}{9}$                  

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.