Probability Distribution Function Examples

Example 1: Suppose we toss two dice. Make a table of the probabilities for the sum of the dice. The possibilities are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Solution:

Probability Distribution Table

X	P(x)
2	1/36
3	2/36
4	3/36
5	4/36
6	5/36
7	6/36
8	5/36
9	4/36
10	3/36
11	2/36
12	1/36

Example 2: The number of old people living in houses on a randomly selected city block is described by the following probability distribution.

What is the probability that 6 or more old peoples live in a randomly selected house?

Solution:

Sum of all the p(probability) is equal to 1

Probability that six or more old peoples live in a house,

= 1 – (0.50 + 0.25 + 0.10)

= 0.15

Thus, probability that six or more old peoples live in a house is equal to 0.15

Example 3: When a fair coin is tossed 8 times, Probability of:

• Exactly Four Heads
• At least Four Heads

Solution:

Every coin tossed can be considered as the Bernoulli trial. Suppose X be the number of heads in this experiment,

n = 8

p = 1/2

So,

P(X = x) = ⁿC_x p^{n – x} (1 – p)^x, x =  0, 1, 2, 3,…n

P(X = x) = ⁸C_xp^{8 – x}(1 – p)^x

P(Exactly 4 Heads)

= P(x = 4)

= ⁸C₄ p⁴ (1 – p)⁴

= 8!/4!4!(1/2)⁴(1/2)⁴

= (8 × 7 × 6 × 5/2 × 3 × 4) × (1/16) × (1/16)

= 420/1536

= 35/128

Thus, the probability of Exactly Four Heads in a Eight Coin Toss experiment is 35/128

P(At Least 4 Heads)

= P(X >= 4)

= P(X = 4) + P(X = 5) + P(X = 6)+ P(X = 7) + P(X = 8).

= ⁸C₄ p⁴ (1 – p)⁴ + ⁸C₅ p³ (1 – p)⁵ + ⁸C₆ p² (1 – p)⁶ + ⁸C₇ p¹(1 – p)⁷ + ⁸C₈(1 – p)⁸

= 8!/4!4!(1/2)⁸ + 8!/5!3!(1/2)⁸ + 8!/6!2!(1/2)⁸ + 8!/7!1!(1/2)⁸ + 8!/8!(1/2)⁸

= 8 × 7 × 6 × 5/4 × 3 × 2 × 256 + 8 × 7 × 6/3 × 2 × 256 + 8/256 + 1/256

= 1680/6144 + 336/1536 + 9/256

= 70/256 +  56/256 + 9/256

= 135/256

Thus, the probability of Atleast Four Heads in a Eight Coin Toss experiment is 135/256

Example 4: Calculate the probability of getting 10 heads, if a coin is tossed 12 times.

Solution:

Given,

• Number of Trials(n) = 12
• Number of Success(r) = 10 (getting 10 heads)
• Probability of Single Head(p) = 1/2 = 0.5

To find ⁿC_r =  n!/r!(n – r)!

=  12!/10!(12 – 10)!

=  (12 × 11 × 10!)/10!2!

= 66

To find p^r = (0.5)¹⁰ = 0.00097665625

So, the probability of getting 10 heads is:

P(x) = ⁿC_r p^r (1 – p)^{n – r}

= 66 × 0.00097665625 × (1 – 0.5)^(12-10)

= 0.0644593125 × (0.5)²

= 0.016114828125

The probability of getting 10 heads = 0.0161…

Example 5: Suppose that each time you take a free throw shot, you have a 35% chance of making it. If you take 25 shots, what is the probability of making exactly 15 of them?

Solution:

Given,

• n = 25
• r = 15
• p = 0.35
• q = 0.65

Compute

C_25,15 (0.35)¹⁵ (0.65)¹⁰ = 0.165

There is a 16.5% chance of making exactly 15 shots.

Example 6: There is a total of 5 people in the room, what is the possibility that someone in the room shares His / Her birthday with at least someone else?

Solution:

P(s) = p(At least someone shares with someone else)                        

P(d) = p(No one share their birthday everyone has a different birthday)

p(s) + p(d) = 1 or 100%

p(s) =100% – p(d)

There are 5 people in the room, the possibility that no one shares his/her birthday

= (365 × 364 × 363 × 362 × 361) ⁄ (365)⁵

= (365! ⁄ (365 – 5)!) ⁄ 365⁵

= (365! ⁄ 360!) ⁄ 365⁵

= 0.9728

p(d) = 0,9728 or 97.28%

p(s) = 100% – p(d)

= 100% – 97.28% or 1 – 0.9728

= 2.72%  ≈ 0.0272

Probability Distribution Function is a function in mathematics that gives the probability of all the possible outcomes of any event. We define probability distribution as how all the possible probabilities of any event are allocated over the distinct values for an unexpected variable.

Two functions are used to describe the Probability Distribution of the function which includes, Probability Mass Function and Probability Distribution Function. Here in this article, we will learn about the Probability Distribution Function. There are two ways to represent the data, Discrete Data and Continuous Data, and based on that we can represent the Probability Distribution Function into two further categories.

In this article, we will learn about Probability Distribution, Probability Distribution Function, its Formula, Graphs, related Examples, and others in detail.