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## Theory of Probability

** Lets try out some examples on the Theory of Probability **

** Example 1 - If we toss the coin 1 time then what is the probability of getting a Head and Tail ? **

Solution - The steps are as follows -

Let E is the event of getting a Head

Let F is the event of getting a Tail

The number of possible outcome are : 2 (i.e. Head and tail)

If we toss the coin 1 time then

The number of possible outcome to E (i.e. getting Head) is : 1

The number of possible outcome to F (i.e. getting Tail) is : 1

Now let's apply the formula

** Example 2 - A bag contains ball of same sizes red, green and yellow. If we take out one ball from bag without looking into it then what is the probability of taking out red ball, green ball and yellow ball ? **

Solution -

Let R is the event of getting red ball

Let G is the event of getting green ball

Let Y is the event of getting yellow ball

The number of possible outcomes are : 3 (i.e. Red, Green and Yellow Ball)

If we take out 1 ball from bag then

The number of possible outcome to R (i.e. getting Red) : 1

The number of possible outcome to G (i.e. getting Green) : 1

The number of possible outcome to Y (i.e. getting Yellow) : 1

Now let's apply the formula

** Example 3 - If we throw a dice once then find**

1) what is the probability of getting number greater than 4

2) what is the probability of getting number less than and equal to 4

Solution -

Let E is the event of getting number > 4

Let F is the event of getting number <= 4

The number of possible outcomes are : 6 (i.e. 1,2,3,4,5,6)

If we throw the device once then

The number of possible outcomes to E (getting number > 4) : 2 (i.e. number 5 and 6)

The number of possible outcomes to F (getting number <= 4) : 4 (i.e. number 1,2,3 and 4)

Now let's apply the formula

As E, F in this example are not same these this are called non elementary events and is denoted by sign ⋶ (not E);

** Example 4 - If A and B are playing and probability of winning the match for A is 0.65. Find the probability of B winning the match **

Solution: Steps are as follows -

Let E is the event of A winning the match

Let F is the event of B winning the match

The number of possible outcome of match is : 1

Possibility of A wining the match is 0.65 (given)

Possibility of B wining the match is : 1 - 0.65 = 0.35

** Theory of Probability ** is used to find the results of random experiments or analysis done on a set of data. Though these experiments only provide an estimate of the probabilities which we can get from the data, the results of data may be different each time we try it with different probability estimates.

Formula of probability is -

P(Event) = | Number of outcomes favorable to Event Number of all possible outcomes of the experiment |

Solution - The steps are as follows -

Let E is the event of getting a Head

Let F is the event of getting a Tail

The number of possible outcome are : 2 (i.e. Head and tail)

If we toss the coin 1 time then

The number of possible outcome to E (i.e. getting Head) is : 1

The number of possible outcome to F (i.e. getting Tail) is : 1

Now let's apply the formula

P(Event) = | Number of outcomes favorable to Event Number of all possible outcomes of the experiment |

P(E) i.e. Head = | 12 |

P(F) i.e. Tail = | 12 |

Solution -

Let R is the event of getting red ball

Let G is the event of getting green ball

Let Y is the event of getting yellow ball

The number of possible outcomes are : 3 (i.e. Red, Green and Yellow Ball)

If we take out 1 ball from bag then

The number of possible outcome to R (i.e. getting Red) : 1

The number of possible outcome to G (i.e. getting Green) : 1

The number of possible outcome to Y (i.e. getting Yellow) : 1

Now let's apply the formula

P(Event) = | Number of outcomes favorable to Event Number of all possible outcomes of the experiment |

P(R) i.e. Red Ball = | 13 |

P(G) i.e. Green Ball = | 13 |

P(Y) i.e. Yellow Ball = | 13 |

1) what is the probability of getting number greater than 4

2) what is the probability of getting number less than and equal to 4

Solution -

Let E is the event of getting number > 4

Let F is the event of getting number <= 4

The number of possible outcomes are : 6 (i.e. 1,2,3,4,5,6)

If we throw the device once then

The number of possible outcomes to E (getting number > 4) : 2 (i.e. number 5 and 6)

The number of possible outcomes to F (getting number <= 4) : 4 (i.e. number 1,2,3 and 4)

Now let's apply the formula

P(Event) = | Number of outcomes favorable to Event Number of all possible outcomes of the experiment |

P(E) i.e. Number > 4 = | 26 | = | 1 2 |

P(F) i.e. Number < 4 = | 4 6 | = | 2 3 |

As E, F in this example are not same these this are called non elementary events and is denoted by sign ⋶ (not E);

Solution: Steps are as follows -

Let E is the event of A winning the match

Let F is the event of B winning the match

The number of possible outcome of match is : 1

Possibility of A wining the match is 0.65 (given)

Possibility of B wining the match is : 1 - 0.65 = 0.35