probability basics
TRANSCRIPT
BASIC TERMS
Probability: Measure if likeliness than an event will occur.
Sample Space: Set of all possible outcomes.
Event: Any subset of any outcomes of any experiment.
Mutually exclusive event: Event which cannot occur together.
Dependent and Independent events.
PRACTICE – 1
A dice is thrown, find probability
P(even no) =
P(no less than 3) =
P(not 3) =
In a deck of 52 cards
P(ace) =
P(red card) =
P(face card) =
RULE OF ADDITION
Mutually exclusive event:
Not Mutually exclusive event:
)()()( BPAPAorBP
)()()()( BandAPBPAPAorBP
PRACTICE – 2
A Dice is thrown once: P( 2 or 3) =
P(multiple of 2 or multiple of 3) =
In Deck of 52 cards P(red or face) =
One ball drawn: Bag contains – 10 Red, 15 Yellow, 25 Green P( Yellow or Green) =
RULE OF MULTIPLICATION
Independent Events:
Dependent Event:
)()()( BPAPBandAP
)|()()( ABPAPBandAP )|()()( BAPBPBandAP Or
PRACTICE – 3
Bag contains – 10 Red, 15 Yellow, 25 Green
Two balls drawn (with replacement): P(1st Red, 2nd Yellow) = P(Both Red) = P(One Red, one Yellow) =
Two balls drawn (without replacement): P(1st Red, 2nd Yellow) = P(Both Red) = P(One Red, one Yellow) =
COIN
Two unbiased coins are tossed simultaneously:
P(one head) =
P(atleast one head) =
P(No head) =
P(one Head one Tail) =
COIN
Three unbiased coins are tossed simultaneously:
P(two head) =
P(at least two head) =
P(at most two head) =
P(All head) =
PERMUTATION & COMBINATION Permutation: “one or more elements of a set where order does matter”
Combination: “one or more elements of a set where order does not matter”
For e.g.: Take 3 Letters – ABCPermutation of 2 of these letter: 3P2 = 6AB BAAC CABC CB
Combination of 2 of these letter: 3C2 = 3ABACBC
Now Take 4 : Letter – ABCDAnd write Permutation & Combination of 3 of these.
P&C
Permutation
Combination
3𝐶 2=3 !
(3−2 )! 2!=3n𝐶𝑟=
𝑛 !(𝑛−𝑟 )!𝑟 !
= 6n𝑃 𝑟=𝑛 !
(𝑛−𝑟 )!
*We will focus majorly on COMBINATION
PRACTICE – 4
Bag is Back : 10 Red, 15 Yellow, 25 Green
Three bolls drawn at random, find:
P(each different color) =
P(2 red 1 green) =
P(exactly 2 green) =
P(at least 2 green) =
P(at most 2 green) =
PRACTICE – 4
Bag is Back : 10 Red, 15 Yellow, 25 Green
Two draws of three balls each are drawn at random (with replacement)
P(1st three Red, 2nd three yellow) =
P(all different color, all different color) =
P(one draws three red, other draws three yellow) =*Wanna try without replacement??
BAYES’ THEOREM
What is the probability of outcome A given that outcome B has already occurred. i.e. P(A|B)
)'().'|()().|(
)().|()|(
APABPAPABP
APABPBAP
n. , . . . 2, 1, =k ,)B|P(A × )P(B
)B|P(A × )P(B = A)|P(B
1ii
ikk
n
i
Believe me! You don’t need to remember this, if you Still go with your Probability Mantra and your logics!
ILLUSTRATION
Late
BusCar Train
P(train)train)|P(lateP(bus)bus)|P(lateP(car)car)|P(late
P(car)car)|P(late late)|P(car
P(Car) = P(Bus) = P(Train) = 1/3
P(late|car) = 1/6P(late|bus) = 1/5P(late|train) = 1/4
Given that You are Late, what is the probability that you travelled by car??
PRACTICE – 5
A bulb is manufactured in one of the three machines X,Y,Z.
The machines produce in capacity: X – 30%, Y – 30%, Z – 40%.
Probability of bulb found defective in machine X – 10%, Y – 15%, Z – 20%.
A bulb is drawn at random and is found to be defective.
What is the probability that the bulb was produced in Machine Y??