PCC | Mahmud | Statistics| statmania.info |

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

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

Solution

There are 6 numbers: 1, 2, 3, 4, 5, 6

There are 3 odd numbers: 1, 3, 5

- \(\therefore P = \frac 3 6 = \frac 1 2\)
- What if the die is unfair?

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

Whenever we have `or`

, we have to add probabilities, if mutually exclusive.

- What if not mutually exclusive?

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

- The Head appears at the first draw?
- At least one Head appears?
- Less than two Heads appear?
- Only Tails appears?

First Two Flips | |||||
---|---|---|---|---|---|

HH | HT | TH | TT | ||

Third Flip | H | HHH | HHT | HTH | HTT |

T | THH | THT | TTH | TTT |

What is the probability that

- All three are Heads?
- There are more than one Head?
- The second draw gives a Head?
- The third draw does not give a head?
- The first draw gives a Tail but the the Draw does not?
- At most one Head appears?

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

- The first numbers is odd?
- The summation of numbers in two draws is a prime number?
- Both numbers are same?
- The second number is bigger?

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

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

When to add when to multiply?

Usually

`and`

implies multiplication (\(\cap\))`or`

implies addition (mutually exclusive) (\(\cup\))

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

\(\frac 3 9 \times \frac 4 8\)

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 |

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

- k = ?
- Find probability of each value of x
- 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

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