Chapter 17: PROBABILITY


Introduction




Sample Space: It is a set of all possible outcomes of an experiment i.e. a sample space for an experiment of rolling a die is {1,2,3,4,5,6} and tossing a coin = {H,T}.

Event: It is a subset of a sample space i.e. Heads or Tails.

Mutually exclusive events: If two or more events have nothing in common then they are mutually exclusive.

The sum of probabilities is 1 i.e The probability of getting heads on a coin toss is 1/2 and getting tails on a coin toss is 1/2 so total probability of all events in a sample space is 1.

Conditional probability: P ( B / A ) is the probability of an event B occurring given that event A has already occurred.



P ( B / A ) = P (A ∩ B) / P (A)
P (A ∩ B) = P (A) * P ( B / A ) 

Independent events:
If the happening of an event is not influenced by another event then we say A and B are mutually independent. Mutually exclusive events in general are not independent.
 P (A ∩ B) = P (A) * P ( B )

P (atleast one) = 1 - P (none)

Problems and Solutions




Q. Two unbiased coins are tossed find probability of getting one head ?

A. Sample space = { HH , TT , HT , TH }

P ( atmost one head ) = 3/4 - include case zero heads and 1 heads

P ( atleast one tail) = 3/4 - include case 1 tails and 2 tails

P ( no tails) = 1 / 4


Q. Find the probability of getting sum of faces more than 6 when 2 dies are rolled?

A. Sample space = { (1,1)     (1,2)     (1,3)     (1,4)     (1,5)     (1,6)
(2,1)     (2,2)     (2,3)     (2,4)     (2,5)     (2,6)
(3,1)     (3,2)     (3,3)     (3,4)     (3,5)     (3,6)
(4,1)     (4,2)     (4,3)     (4,4)     (4,5)     (4,6)
(5,1)     (5,2)     (5,3)     (5,4)     (5,5)     (5,6)
(6,1)     (6,2)     (6,3)     (6,4)     (6,5)     (6,6) }


P ( sum more than 6 ) = 21 / 36

P ( sum is multiple of 3 ) = 11 / 36




Q. A bag contains 6 white balls and 4 black balls. Probability that they are of same color.

A. P (event) =

= P (choosing 2 white balls from 6) + P( choosing 2 black from 4)

= (6C2/10C2) + (4C2 / 10C2)


Q. Probability that 2 cards drawn at random from pack of cards are both black, both red , both face cards [queen, king, jack], one face card and other normal, both are aces.

A. n ( sample space ) = sample space size is 52.

P ( both black cards ) = number (black cards) / number (cards) = 26C2/52C2 = 1/2

P ( both red cards ) = 1/2

P ( both face cards) = 12 / 52

P ( one face card and other normal ) = (12C1 * 40C1 ) / 52C2.

P ( both aces ) = 4C2 / 52C2.



Q.There are ten 50 paise coins placed on a table. Six of these show tails, four show heads. A coin is chosen at random and flipped over (not tossed). This operation is performed seven times. One of the coins is then covered. Of the remaining nine coins, five show tails and four show heads. The covered coin shows


  1. a head
  2. a tail
  3. more likely a head
  4. more likely a tail

Ans.a



Quiz

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