# Probability

Probability

Probability may be defined as the measure of the chance of the event that wiil occur as a result of t he experiment.

Mutually exclusive events: Two events are said to be mutually exclusive event if they do not occur at the same time. If A and B are the two events of the experiment and S be the sample space

If P (AUB)= P (A) + P (B) where P (A) = $\frac{{n\left( A \right)}}{{n\left( S \right)}}$,P (B) = $\frac{{n\left( B \right)}}{{n\left( S \right)}}$

Dependent and independent events

Two or more events are said to be independent event if occurrence of one event doesn’t affect the occurrence of other.

If two event are said to be dependents if P(A ∩ B) = P ( A) ×P ( B)

Two or more events are said to be dependent event if occurrence of one event affect the occurrence of other.

P (A ∩ B) = P (A) × P(B/A) If event A has already occurred.

Probability Tree diagram

It’s a probability diagramrepresenting all the possible outcomes of the experiments.

Example 1

A bag contains 3 red balls and 4 blue balls .If the two balls are drawn in succession without replacement. Show the probability in tree diagram.

Example 2

What is the probability of getting a card having cube number or a prime number when a card is drawn randomly from a number cards, numbered from8 to 28?Find it. s

soln

Cube number from 8 to 28C = { 8, 27} ,n(C ) = 2

Prime number from 8 to 28P = (11, 13, 17, 19, 23,}, n (P)= 5

Total number n(S) = 21

Probability of getting cube number P(C) = $\frac{{{\rm{n}}\left( {{\rm{C\: }}} \right)}}{{{\rm{\: n}}\left( {\rm{S}} \right){\rm{\: }}}}{\rm{\: }}$= $\frac{2}{{21}}$

Probability of getting prime number P(P) = $\frac{{{\rm{n}}\left( {\rm{P}} \right)}}{{{\rm{\: n}}\left( {\rm{S}} \right){\rm{\: }}}}{\rm{\: }}$=$\frac{5}{{21}}$

Probability of getting prime number or cube number = $\frac{{{\rm{n}}\left( {\rm{P}} \right)}}{{{\rm{\: n}}\left( {\rm{S}} \right){\rm{\: }}}} + \frac{{{\rm{n}}\left( {{\rm{C\: }}} \right)}}{{{\rm{\: n}}\left( {\rm{S}} \right){\rm{\: }}}}{\rm{\: }}$= $\frac{2}{{21}} + \frac{5}{{21}}$ = $\frac{7}{{21}}$ = $\frac{1}{3}$

Example 3

A card is drawn randomly from a well shuffled pack of 52cards.what is the probability that the drawn card is either an ace or a face card? Find it.

soln

Let P be the probability A

Number of ace card = 4

Number of face card = 12

P (A) = $\frac{4}{{52}}$

P (F) = $\frac{{12}}{{52}}$

P(A or F)= $\frac{4}{{52}}\:$+ $\frac{{12}}{{52}}$=$\frac{4}{{13}}$

Example 4

From a bag containing 4 rd and 7 blue balls of same size, two balls are drawn randomly in succession without replacement .Show the probabilities of all outcomes on a tree diagram.

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