**Introduction to standard deviation probability help: **

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

Standard Deviation is defined as the calculation of relating the variability of the Data set. The standard deviation is also called as “Root Mean Square Deviation” because it is the square root of the arithmetic mean of the squares of the deviation. The standard deviation is denoted as (s) sigma.

**Ex 1:**

**Calculate the probability of occurrence of getting 2,4 numbers when a dice is rolled?**

**Sol:**

Where,

there are six different outcomes 1, 2, 3,4,5,6

n (A) = 2, total number of 2,4 occurred is 2.

n (S) = 6, total number of outcomes is 6.

Probability of the event occurs = P (A) = n (A) / n(S) = 2 / 6 = ` 1/3` .

Probability of the event does not occur = P (A') = 1 – P (A) = 1 - 1/3 = `2/3` .

**Ex 2:**

**What is the probability of getting a diamond card in a fresh pack of cards?**

**Sol:**

The event is getting a diamond card: There are 13 diamond cards in a pack of 52 cards.

Total sample spaces are 52 cards.

Probability of getting a diamond cards = number of diamond cards / sample space

= 13/52

= 1/4

Probability of getting a black cards = **0.25**

**Ex 3:**

**What is the probability of getting a red king in a pack of 52 cards?**

**Sol:**

In a pack of 52 cards, there are 2 red king.

Total Sample space = 52

Probability of getting a red king = number of red king / total sample space

= 2/52

= 1/26

= **0.04.**

**Ex 4:**

Find the standard deviation for the given values

X = 5,8,10,3,4,7,8,9

**Sol:**

Mean (M) = 5+8+10+3+4+7+8+11 / 8 = 56 / 8 = 7.

(X-M) = 5- 7, 8-7, 10-7, 3-7, 4-7, 7-7, 8-7, 11-7 .

= -2 , 1 , 3 , -4 , -3 , 0 , 1 , 4 .

(X-M)^{2} = 4 , 1 , 9 , 16 , 9 , 0 , 1 , 16

Sum of (X-M)^{2} = 4+1+9+16+9+0+1+16 = 56

Here number of terms N = 8,

N-1 = 8 – 1 = 7.

Now use the standard deviation formula,

SD(σ) = sqrt((X-M)^{2}/(n-1)) = √(56/7) = √8 = 2.828.

**Ex 5:**

Find the standard deviation for the given values

X = 8,5,2,3,4,7,8,9

**Sol:**

Mean (M) = 8+5+2+3+4+7+8+11 / 8 = 48 / 8 = 6.

(X-M) = 8- 6, 5-6, 2-6, 3-6, 4-6, 7-6, 8-6, 11-6 .

= 2 , -1 , -4 , -3 , -2 , 1 , 2 , 5 .

(X-M)^{2} = 4 , 1 , 16 , 9 , 4 , 1 , 4 , 25

Sum of (X-M)^{2} = 4+1+16+9+4+1+4+25 = 64

Here number of terms N = 8,

N-1 = 8 – 1 = 7.

Now use the standard deviation formula,

SD(σ) = sqrt((X-M)^{2}/(n-1)) = √(64/7) = √9.142 = 3.0235.