Probability

Abdullah Al Mahmud

Terms

  • Random Experiment
  • Event
  • Equally Likely
  • Mutually Exclusive
  • Sample Space
  • Sample Point
  • Unbiased/fair coin/die

Example 01

What is the probability, if a fair die is thrown, that an odd number is found?

Solution

Example 02

There are 4 red, 6 white balls, and 5 black balls in an urn. If a ball is picked randomly what is the probability that it is red or white?

Solution

Coin And Die Problem

Tossing A Coing Twice

First Coin
H T
Second
Coin 		H HH HT
T TH TT

Tossing a coin twice is equivalent to tossing two coins at once

What is the probability that

  1. The Head appears at the first draw?
  2. At least one Head appears?
  3. Less than two Heads appear?
  4. Only Tails appears?

Flipping A Coin Thrice

First Two Flips
HH HT TH TT
Third Flip H HHH HHT HTH HTT
T THH THT TTH TTT

What is the probability that

  1. All three are Heads?
  2. There are more than one Head?
  3. The second draw gives a Head?
  4. The third draw does not give a head?
  5. The first draw gives a Tail but the the Draw does not?
  6. At most one Head appears?

Flinging Two Dice at Once

Tossing Two
Dice at Once 		First Die
1 2 3 4 5 6
Second
Die 		1 1,1 1,2 1,3 1,4 1,5 1,6
2 2,1 2,2 2,3 2,4 2,5 2,6
3 3,1 3,2 3,3 3,4 3,5 3,6
4 4,1 4,2 4,3 4,4 4,5 4,6
5 5,1 5,2 5,3 5,4 5,5 5,6
6 6,1 6,2 6,3 6,4 6,5 6,6

What is the probability that

  1. The first numbers is odd?
  2. The summation of numbers in two draws is a prime number?
  3. Both numbers are same?
  4. The second number is bigger?

Example 03

The probability that a person travels from Dhaka to Khulna by bus is \(\frac 2 5\), that from Khulna to Rajshahi by train is \(\frac 5 8\).

  1. Make a probability tree
  2. Find the probability that he would travel to khulna by bus and to Rajshahi not by bus.

Addition or Multiplication?

When to add when to multiply?

Usually

  • and implies multiplication (\(\cap\))
  • or implies addition (mutually exclusive) (\(\cup\))

And/Or Example

There are 3 red balls, 2 blue balls and 4 white balls in a box.

  • package technique

If a ball is selected at random, what is the probability that it is red or blue?

  • \(P(R \cup B)=P(R)+P(B)\)

If two balls are drawn without replacement, what is the probability that the first one is red and the second one is white?

Solution

Excercise 11

The employees in a factory can be classified based on works they perform, as mentioned below:

Classification # Employees
Managerial 157
Inspection 52
Production 1473
Other 215

Probability Function

Given, \(P(x) = \frac{2x+k}{56}; x = -3, -2, -1, 0, 1, 2, 3\)

  1. k = ?
  2. Find probability of each value of x
  3. Find \(P(-2 \le x \le 2)\)

Clue: If \(S=\{H,T\}, P(H) + P(T) = 1\),

i.e. \(\sum\) P(All possible values) = 1

Analytic Problem

Arrange 10 soldiers in 5 rows with 4 in each row.