Probability of Rolling Doubles

Probability is the chance of the outcome of an event of a particular experiment. Probabilities are occurs always numbers between 0(impossible) and 1(possible). The set of all possible outcomes of a particular experiment is called as sample space. For example probability of getting a 3 when rolling a dice is ` 1/6` . In this article we will discuss about probability problems using dice.


Rules of Probability Doubles - Example Problems

Example 1: If rolling two dice, what is the probability of getting doubles?

Solution:

Let S be the sample space, n(S) = 6 * 6 = 36.

A be the event of getting doubles.

A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, n(A) = 6

P(A) = `(n(A))/(n(S))` = `6/36` = `1/6`

Example 2: If rolling two dice, what is the probability of getting doubles or primes?

Solution:

Let S be the sample space, n(S) = 6 * 6 = 36.

A be the event of getting doubles.

n(A) = {{1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} = 6

P(A) = `(n(A))/(n(S))` = `6/36` = `1/6`

Let B be the event of getting primes.

n(B) = {(1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4, 3), (5,2), (5, 6), (6, 1), (6, 5)} = 15

P(B) = `(n(B))/(n(S))` = `15/36` = `5/12`

P(A or B) = P(A) + P(B) = `1/6` + `5/12` = `7/12`

P(A or B) = `7/12`

Therefore probability of getting doubles or primes is `7/12`

Example 3: If rolling two number cubes, what is the probability of getting doubles or sum of 7?

Let S be the sample space, n(S) = 6 * 6 = 36.

A be the event of getting doubles.

n(A) = {{1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} = 6

P(A) = `(n(A))/(n(S))` = `6/36` = `1/6`

Let B be the event of getting sum of 7.

n(B) = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)} = 6

P(B) = `(n(B))/(n(S))` = `6/36` = `1/6`

P(A or B) = P(A) + P(B) = `1/6` + `1/6` = `1/3`

P(A or B) = `1/3`

Therefore probability of getting doubles or sum of 7 is `1/3`

Probability of Rolling Doubles - Practice Problems

Problem 1: If rolling two dice, what is the probability of getting a sum of 5 or 6?

Problem 2: If rolling two number cubes, what is the probability of getting 6 or 7?

Answer: 1) `1/4` 2) `11/36`

Solving Double Number Identities

Trigonometry is arrived from the Greek word, trigonon = triangle and metron = measure. The father of trigonometry is Hipparchus. He designed the first trigonometric table. Identity is defined as an equation that is true for all probable values of its variables. In online, many websites provide online tutoring using tutors. Double number trigonometric identities problems are easy to solve. Solving double number trigonometric identities problems are easy.  Through practice, students can learn about solving trigonometric identities. Through online, students can practice more problems on trigonometric identities. In this topic, we are going to see about; solving double number identities.

Solving Double Number Identities: - Double Number Identities

The list of double number identities are given below,

sin `2theta` = 2sin `theta` cos `theta `

cos `2theta` = 2cos2 `theta` -1

cos `2theta` = 1- 2sin2 `theta`

cos `2theta` = cos2 `theta` – sin2 `theta`


Solving Double Number Identities: - Examples

Example 1:

Evaluate sin 60 using double number identities,

Solution:

sin 2theta = 2sin theta cos theta

sin 60 = (2x30)

= 2 sin 30 cos 30

= 2 (0.5) (0.866)

= 2*0.433

= 0.866

The answer is 0.866



Example 2:

Evaluate sin 90 using double number identities,

Solution:

sin 2theta = 2sin theta cos theta

sin 90 = (2x45)

= 2 sin 45 cos 45

= 2 (0.707) (0.707)

= 2*0.499

= 1

The answer is 1



Example 3:

Evaluate cos 50 using double number identities,

Solution:

cos 2theta = cos2 theta – sin2 theta

cos 50 = cos (2x25)

= cos2 25 - sin2 25

= (0.906)2 - (0.423)2

= 0.820 - 0.179

= 0.641

The answer is 0.641

Example 4:

Evaluate cos 90 using double number identities,

Solution:

cos 2theta = cos2 theta – sin2 theta

cos 90 = cos (2x45)

= cos2 45 - sin2 45

= (0.707)2 - (0.707)2

= 0.499-0.499

= 0

The answer is 0



Example 5:

Find cos 2y if sin y = -15/16 and in 3rd quadrant

Solution:

It is given that sin 2y is in 3rd quadrant,

Use the double angle identities

cos 2theta = 1- 2sin2 theta

cos 2y = 1 -2sin2 y

= 1 – 2`(-15/16)` 2

= 1 – 2 `(225/256)`

Taking LCM, we get

= `(256-450)/256`

= `-194/256`

= -97/128

The answer is `-97/128`