Probability Distribution: Overview, Questions, Preparation

Probability is the measure of the chance of occurrence of any event. A list of all the possible outcomes is known as the sample space. If an event E occurs, its probability is denoted by P(E).
The conditional probability of an event occurring concerning the occurrence of another event is given as-

P(M|N)= (Number of events favourable to M N) / (Number of events favourable to N)
= n (M N) / n(N).

Random Variable:

A random variable is a real numbered entity. The domain of a random variable is the sample space of a random experiment.

A descriptive depiction of the values of the random variable along with their associated probabilities is known as the probability distribution of that random variable.

Probability Distribution:

The probability of a random variable V is given as -
V : v1, v2, v3, ... , vn
P(V): p1, p2, p3, …, p_n

So, the probability distribution lists all possible variables and the outcome probabilities. The sum of all probabilities in a probability distribution must be one.

The topic “probability distribution” is a part of the chapter “Probability” in the class 12 mathematics syllabus. This topic carries a weightage of around 7-8 marks in the examination.

 1. A discrete random variable X has the following probability distribution:

Find the value of K. Also, calculate the mean.

Solution:

We know that,

So, K + 2K+ 2K + 3K + K² + 2K² + 7(7K² +K) = 1

=> 10K² + 9K- 1= 0 => (10K -1)(K+1) =0

=> K= 1/10 or K= -1

Since probability cannot be negative, K= 1/10.

Mean=

⁼  

=> Mean = 1 x K + 2x 2K + 3x 2K + 4x 3K + 5x K² + 6x 2K² + 7x 7K²

= 1 x (1/10) + 2x 2(1/10) + 3x 2(1/10) + 4x 3(1/10) + 5x (1/10)² + 6x 2(1/10)² + 7x {7(1/10)² + 1/10}

= 1/10 + 4/10 + 6/10 + 12/10 + 5/100 + 12/100 + 49/100+ 7/10

Mean= 3.66

2.Show the probability distribution of the number of heads in three tosses of a coin.

Solution:

Let D denote the number of heads tossed. So, D can take the values 0, 1, 2, 3. When a coin is tossed three times, we get

Sample Space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

P (D = 0) = P (no head) = P (TTT)= ⅛

P (D = 1) = P (one head) = P (HTT, THT, TTH) = ⅜ 

P (D = 2) = P (two heads) = P (HHT, HTH, THH) = ⅜ 

P (D = 3) = P (three heads) = P (HHH)= ⅛ 

Hence, the probability distribution of D is given as-

Q: What are the conditions of the probability distribution?

Q: What are the types of the probability distribution?

Q: What are some examples of the normal probability distribution?

Q: What is a binomial probability distribution?

Q: What is the Poisson probability distribution?