A bag contain 4 red, 5 blue and 3 green balls if two ball are drawn at random. What is the probability that both are red? 

A bag contain 4 red, 5 blue and 3 green balls if two ball are drawn at random. What is the probability that both are red?

The probability of drawing both balls as red is 1/11.

To find the probability of drawing two red balls from the bag, we first need to find the total number of ways to draw two balls from the bag, and then find the number of ways to draw two red balls.

Total number of ways to draw two balls from the bag:

There are 12 balls in total. So, the total number of ways to draw two balls without any restrictions is given by the combination formula:

Total number of ways = 12 choose 2

Total number of ways = (12 * 11) / (2 * 1)

Total number of ways = 66

Number of ways to draw two red balls:

Number of ways to draw two red balls = 4 choose 2

Number of ways to draw two red balls = (4 * 3) / (2 * 1)

Number of ways to draw two red balls = 6

Therefore, the probability of drawing two red balls is:

P(both red) = Number of ways to draw two red balls / Total number of ways to draw two balls

P(both red) = 6 / 66

P(both red) = 1 / 11

So, the probability of drawing both balls as red is 1/11.

What is the Probability Theory?

Probability theory is a branch of mathematics that deals with the analysis of random phenomena. It provides a framework for quantifying uncertainty and making predictions based on this uncertainty. The fundamental concept in probability theory is the notion of probability, which represents the likelihood of different outcomes occurring in an uncertain situation.

Key components of probability theory include:

1. Sample Space: This is the set of all possible outcomes of a random experiment.

2. Events: An event is a subset of the sample space, representing a particular outcome or combination of outcomes.

3. Probability Measure: A function that assigns a numerical value between 0 and 1 to each event, representing the likelihood of that event occurring.

4. Probability Distribution: Describes how the probabilities are distributed among the different possible outcomes.

5. Random Variables: These are variables that take on different values as a result of a random experiment. Probability distributions can be used to describe the likelihood of each possible value of a random variable.

6. Expected Value: Also known as the mean or average, it represents the long-term average value of a random variable, weighted by its probabilities.

7. Variance and Standard Deviation: Measures of the spread or dispersion of a probability distribution.

8. Independence and Dependence: Events are considered independent if the occurrence of one does not affect the occurrence of another, and dependent otherwise.

Probability theory is widely applied in various fields including statistics, finance, physics, engineering, computer science, and more. It provides a powerful framework for reasoning about uncertainty and making informed decisions in the face of randomness.