MA 485-585 PROBABILITY THEORY QUIZ #1 JANUARY 22 2. An urn contains 7 black and 4...
MA 485-585 PROBABILITY THEORY QUIZ #1 JANUARY 22 Time alotted: 15 min. Each question is 10 points. 1. We flip a fair coin 7 times. What is the conditional probability of the event that we get exactly 1 Head given the condition that the number of Heads is at most 3?
2(15)(a) An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. This continues until exactly 2 of the 4 chosen are white. What is the probability that we shall maken selections? (b)Compute E[x2] for a Poisson random variable X.
An urn “A' contains 2 white and 4 black balls. Another urn ®contains 5 white and 7 black ball. A ball is transferred from the urn $A)to urn B'. Then a ball is drawn from urn B'. Find the probability, thatit will be white.
An urn contains 10 white and 6 black balls. Balls are randomly selected, one at a time, until a black one is obtained. If we assume that each ball selected is replaced before the next one is drawn, what is the probability that a) exactly 5 draws are needed? b) at least 3 draws are needed?
4. Urn A contains 3 red and 3 black balls, whereas urn B contains 4 red and 6 black balls. If a ball is randomly selected from each urn, what is the probability that the balls will be the same color?
Urn I contains 5 white balls and 5 black balls. Urn II contains 7 white balls and 3 black balls. Under which of the following plans is the probability of getting two white balls the greatest? (a) Draw one ball from each urn. (b) Draw two balls from Urn I. (c) Put all 20 balls in one urn and then draw two.
3.14. An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of (a) the first 2 balls selected are black and the next 2 are white (b) of the first 4 balls selected, exactly 2 are black.
An urn contains M white and N black balls. Balls are randomly selected, one at a time, until a black one is obtained. If we assume that each ball selected is replaced before the next one is drawn, what is the probability that a) exactly x draws are needed? b) at least k draws are needed?
An urn contains 7 white balls and 3 black balls. Two balls are selected at random without replacement. What is the probability :1 The first ball is black and the second ball is white.? 2: One ball is white and the other is black? 3:the two balls are white ?
Urn "A" contains 5 white balls and 4 black balls, whereas urn B contains 3 white balls and 5 black balls. A ball is drawn at random from urn "B" and placed in urn "A". A ball is then drawn from urn "A". It happens to be black. What is the probability that the ball transferred was black?