What is the probability of getting 20 points with 6 dice?

Probability is a likelihood of an event occurring. It is a number that comes in between 0 and 1, where 0 means that the event is impossible and 1 means that the event is certain to occur.

The probability of an event A is denoted by P(A) and is defined as the ratio of the number of outcomes that correspond to A to the total number of possible outcomes. 

The formula to calculate the probability is discussed below in the image,

For example, if we flip a coin the probability of getting the head is 1/2 as the number of outcomes of the head is 1 and the total number of outcomes is 2.

Formula of Probability

The formula used to calculate the probability of the event is,

Probability of an Event = {Number of ways it can occur} ⁄ {Total number of outcomes}

P(A) = {Number of ways A occurs} ⁄ {Total number of outcomes}

What is the probability of getting 20 points with 6 dice?

Solution:

Let the probability of getting 20 with 6 dice be P.

To find the number of outcomes that result in a total score of 20, we can use a technique called generating functions. The generating function for a single die is (x + x² + x³ + x⁴ + x⁵ + x⁶), since each term represents the probability of rolling a specific value on the die. To find the generating function for 6 dice, we can simply multiply the generating function for a single die by itself 6 times:

(x + x² + x³ + x⁴ + x⁵ + x⁶)⁶

To find the sum 20 we have to find the coefficient of x²⁰ in (x + x² + x³ + x⁴ + x⁵ + x⁶)⁶

Take x common from the equation

Coefficient of x²⁰ in x⁶(1 + x + x² + x³ + x⁴ + x⁵)⁶

Coefficient of x¹⁴ in (1 + x + x² + x³ + x⁴ + x⁵)⁶

Using the sum of GP we get,

Coefficient of x¹⁴ in  [(1-x⁶) / (1-x)]⁶

Coefficient of x¹⁴  in (1 – x⁶)⁶  × (1 – x)^-6 ……(1)

Expansion of (1-x⁶)⁶ = 1 – (⁶C₁)×x⁶ + (⁶C₂)×x¹² – (⁶C₃)×x¹⁸+ …….

Coefficient of x¹⁴ so terms after x¹² will be useless because they are greater than x¹⁴ so we will ignore them.

Expansion of (1-x)^-6 = 1 + 6×x + (6×7×x²)/2! + (6×7×8×x³)/3! + …….

From eq(1)

Coefficient of x¹⁴ in [1 – (⁶C₁)×x⁶ + (⁶C₂)×x¹² – (⁶C₃)×x¹⁸+ …….]×[1 + 6×x + (6×7×x²)/2! + (6×7×8×x³)/3! + …….]

= ¹⁹C₁₄ – ⁶C₁×¹³C₈ + ⁶C₂×⁷C₂

= 4221

Therefore there are 4221 ways to get the sum of 20 in 6 dice

Total No. of ways = 6⁶ =46656

P = 4221/46656

   = 0.0904

Therefore the probability of getting the sum of 20 in 6 dice will be 0.0904

Read More,

• Probability Theory
• Probability Distribution

Solved Examples on Probability

Example 1: Find the probability of getting a sum of 8 when rolling two dice.

Solution:

Let the probability of getting  a sum of 8 is P(A)

Total no. of ways to get a sum of 8 when rolling two dice are,

• (2,6)
• (3,5)
• (4,4)
• (5,3)
• (6,2).

The total number of possible outcomes when rolling two dice is 6 x 6 = 36, since each die has 6 possible outcomes.

Therefore, the probability P(A) = 5/36.

Example 2: Find the probability of rolling at least one 6 when rolling two dice.

Solution:

Let the probability of getting at least one 6 in two dice be P(A)

Total no of ways to get at least one 6 when rolling two dice are,

• (1,6)
• (2,6)
• (3,6)
• (4,6)
• (5,6) 
• (6,1)
• (6,2)
• (6,3)
• (6,4)
• (6,5)
• (6,6)

The total number of possible outcomes when rolling two dice is 6 × 6 = 36

Therefore, 

P(A) = Favourable / Total = 11/36

Example 3: If you flip a coin three times, what is the probability of getting exactly two tails?

Solution:

Let the probability of getting exactly two tails be P(A)

No of ways in which we can get three tails are:

• TTH
• THT
• HTT

Hence, there are total 3  ways to get three tails.

The total number of possible outcomes when flipping a coin three times is 2 × 2 × 2 = 8, since each flip has two possible outcomes.

Therefore, P(A) = Favourable Case/ Total Case

                        = 3/8

FAQs on Probability

Q1: What is probability?

Answer: 

Probability is the measure of the likelihood that an event will occur.

Q2: What is the difference between theoretical and empirical probability?

Answer: 

Theoretical probability is the probability that an event will occur based on mathematical calculations and assumptions, while empirical probability is the probability based on actual observations or data.

Q3: What is the formula for calculating probability?

Answer: 

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Q4: What is the difference between independent and dependent events?

Answer: 

Independent events are events that do not affect each other’s probabilities, while dependent events are events where the outcome of one event affects the outcome of the other event.

Q5: What is the law of large numbers?

Answer: 

The law of large numbers is a principle that states that as the sample size increases, the average of the sample will approach the expected value of the population.

Q6: What is the difference between a random variable and a probability distribution?

Answer: 

A random variable is a variable that takes on random values, while a probability distribution is a function that assigns probabilities to the values of a random variable.

Q7:  What is Bayes’ Theorem?

Answer: 

Bayes’ Theorem is a mathematical formula used to calculate the probability of an event based on prior knowledge of conditions that might be related to the event. It is often used in Bayesian statistics.