Class 12

Math

Algebra

Probability I

An urn contains nine balls of which three are red, four are blue, and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is

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An um contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the um and then a ball is drawn at random. What is the probability that the second ball is red?

If A, B, C are three events associated with a random experiment prove that $P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩C)−P(B∩C)+P(A∩B∩C)$

There are two bags, one of which contains 3 black and 4 white balls, while the other contains 4 black and 3 white balls. A fair die is cast, if the face 1 or 3 turns up, a ball is taken from the first bag, and if any other face turns up a ball is chosen from the second bag. Find the probability of choosing a black ball.

$AandB$ play a series of games which cannot be drawn and $p,q$ are their respective chance of winning a single game. What is the chance that $A$ wins $m$ games before $B$ wins $n$ games?

A coin is tossed three times, consider the following events.A : ‘No head appears’, B: ‘Exactly one head appears’ and C: ‘Atleast two appear’.Do they form a set of mutually exclusive and exhaustive events?

Refer to question 6 above, state true or false: (give reason for your answer)(i) A and B are mutually exclusive.(ii) A and B are mutually exclusive and exhaustive.(iii) $A=B_{prime}$(iv) A and C are mutually exclusive. (v) A and $B_{a}reμtuallyexclusive(vi)$A^ , \displaystyle{B}^{{\quad\text{and}\quad}}{C}{a}{r}{e}\mu{t}{u}{a}{l}{l}{y}{e}{x}{c}{l}{u}{s}{i}{v}{e}{\quad\text{and}\quad}{e}{x}{h}{a}{u}{s}{t}{i}{v}{e}.

If $P(E)=0.05$, what is the probability of not E?

Five cards—the ten, jack, queen, king and ace of diamonds, are well shuffled with their face downwards. One card is then picked up at random.(i) What is the probability that the card is the queen?(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?