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
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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.