What is the probability of getting 20 points with 6 dice?
Last Updated :
21 Dec, 2023
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,
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:
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.
Share your thoughts in the comments
Please Login to comment...