QuizSolver
  • Bank PO
  • CBSE
  • IIT JEE
 
 

Solved Subjective Question on Probability Set 1

Posted on - 04-01-2017

Math

IIT JEE

Example.1

A tosses 2 fair coins and B tosses 3 fair coins. The game is won by the person who throws greater number of heads. In case of a tie, the game is continued under identical rules until someone wins the game. Find the probability of A winning the game.

Solution

For a specific game let Ai­­ (Bi) denote the number of heads obtained by A(B) is i when he tosses two (three) fair coins. A will win a particular game in following ways, A1 and B0 occur; A2 and B0 occur, or A2 and B1 occur. If AW be the event that A wins the specific game, then .

P(AW) = P(A1 Ç B0) + P(A2 Ç B0) + P(A2 Ç B1)

= P(A1)×P(B0) + P(A2)×P(B0) + P(A2)×P(B1)

=

Now game results in a tie if A0 and B0 occur or A1 and B1 occur or A2 and B2 occur. If T be event representing a tie;.


&⇒
P(T) = P(A0)×P(B0) + P(A1) ×P(B1) + P(A2)×P(B2)

=

Now required probability,

= P(AW) + P(T)×P(AW) + (P(T))2×P(AW) + L

=

Example.2

A bag contains n white and n red balls. Pairs of balls are drawn without replacement until the bag is empty. Find the probability of each pair consisting of balls of different colours.

Solution

Total number of ways of drawing the balls

= 2nC2 × 2n-2C2 × 2n-4C2L4C2 × 2C2

=

=

For favourable ways we must draw balls in pairs.

Number of choices for first pair = n×n = n2

Similarly number of choices for second pair

= (n -1) × (n -1) = (n - 1)2

similarly for the remaining pairs.


&⇒
Total favourable ways = n2 × (n -1)2×(n -2)2× L 22×12 = (n!)2


&⇒
Probability of the required event is =

Example.3

A draws a card from a pack of n cards marked 1, 2, .... ,n. The card is replaced in the pack and B draws a card. Find the probability that A draws (i) the same card as B, (ii) a higher card than B.

Solution

A and B can draw the cards in n ´ n ways. .

(i) If A draws the same card as B then number of favourable cases in this is n.

Therefore, required probability =

(ii) If A draws card higher than B then number of favourable cases is
(n - 1) + (n - 2) +L + 3 + 2 + 1.

(As when B draws card number 1 then A can draw any card from 2 to n and so on).

Therefore, required probability =

Example.4

To avoid detection at customs, a traveler has placed six narcotic tablets in a bottle containing nine vitamin pills that are similar in appearance. If the customs official selects three of the tablets at random for analysis, what is the probability that traveler will be arrested for illegal possession of narcotics.

Solution

Let A represents the event that sample taken does not posses any narcotic tablet. .


&⇒
P(A) =


&⇒
Probability that traveler would be arrested =

Example.5

A bag contains ‘W’ white and 3 black balls. Balls are drawn one by one without replacement till all the black balls are drawn. What is the probability that this procedure for drawing the balls will come to an end at the rth draw.

Solution

Procedure of drawing the balls has to end at the rth draw,


&⇒
exactly 2 black balls must have been drawn up to (r -1)th draw.

Now probability of drawing exactly 2 black balls up to (r -1) draws

=

=

At the end of (r -1)th draw, we would be left 1 black and (W -r+3) white balls. .


&⇒
Probability of drawing the black ball at the rth draw =


&⇒
Probability of required event =

=

 
Quadratic Equations - Solved Objective Questions Part 2 for Conceptual Clarity
Quadratic Equations - Solved Objective Questions Part 1 for Conceptual Clarity
Solved Objective Question on Probability Set 2
Solved Objective Question on Probability Set 1
Solved Objective Question on Progression and Series Set 2
Solved Objective Question on Permutations and Combinations Set 3
Solved Objective Question on Permutations and Combinations Set 2
Solved Objective Question on Progression and Series Set 1
Solved Objective Question on Permutations and Combinations Set 1
Quadratic Equations - Solved Subjective Questions Part 4
Quadratic Equations - Solved Subjective Questions Part 2
Quadratic Equations - Solved Subjective Questions Part 3
Quadratic Equations - Solved Subjective Questions Part 1
Solved Subjective Questions on Circle Set 9
Solving Equations Reducible to Quadratic Equations
Theory of Polynomial Equations and Remainder Theorem
Solved Subjective Questions on Circle Set 8
Solving Quadratic Inequalities Using Wavy Curve Methods
Division and Distribution of Objects - Permutation and Combination
Basics of Quadratic Inequality or Inequations

Comments