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- Suppose that colored balls are distributed in three indistinguishable boxes as follow: A box is selected at random from which a ball is drawn at random Find the probability that the ball is white Given the ball is Red, what is the probability that box 2 was selected? a. b. BOX 1 2 RED 2 4 3 GREEN 3 2 4 WHITE 2 3 38. Prove the following properties of conditional probability: a. P(|B) = 1, P (Ø|B) = 0; b. P(A₁ U A₂|B) = P(A₁|B) + P(A₂|B) – P(A₁A₂|B); P(A|B) = 1 - P(AB).Assume you have a typical fair 6-sided die. That is to say, it has sides 1, 2, 3, 4, 5, 6 and each side has probability of 1/6 of occurring. You are to roll the die twice, and record the maximum, M, of the two 6 rolls. If you roll a 2 and a 3 (or a 3 and a 2), then M = 3. If you roll a 4 and a 4, then M = 4. The rolls are independent of each other. a. What is the variance of M, var(M). b. Given that the maximum rolled is a 4, what is the probability that one of the rolls was a 2 ?
- pleease aasnwer all the parts:Assume you have a typical fair 6-sided die. That is to say, it has sides 1, 2, 3, 4, 5, 6 and each side has probability of 1/6 of occurring. You are to roll the die twice, and record the maximum, M, of the two 6 rolls. If you roll a 2 and a 3 (or a 3 and a 2), then M = 3. If you roll a 4 and a 4, then M = 4. The rolls are independent of each other. a. Derive the probability table for M. That is to say P(M = m) for m=1,2,3,4,5,6. b. Find the expected value of M, E(M)*2.100 Show that Theorem 2.6, the additive law of probability, holds for conditional probabilities. That is, if A, B, and C are events such that P(C) > 0, prove that P(A U B|C) = P(A|C) + P(B|C)–P(ANB|C). [Hint: Make use of the distributive law (AUB)NC = (ANC)U(BNC).] The Additive Law of Probability The probability of the union of two events A and B is THEOREM 2.6 P(AUB) = P(A) + P(B) – P(AN B). If A and B are mutually exclusive events, P(AN B) = 0 and P(AU B) = P(A)+ P(B).
- Suppose you have a fair four sided dice showing {1,2,3,4}. You roll the dice and then continues to roll the dice until it shows a number that is less or the same as the first roll. Let X be the value of the dice on the first roll and let N be the number of rolls that is made after the first roll. a. Determine if X and N is independent. b. Determine the probability P(N>1) c. Determine the joint mass function p(x,n)=P(X=x, N=n) of vector (X, N)1.3.18. A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct envelope? Hint: Let C; be the event that the ith letter is in the correct envelope. Expand P(C1 U C2U C3) to determine the probability.The zebra plays a game in which in every turn she randomly chooses a letter from the set {A,B,C}, independently of other turns, and registers the outcome in a list. 1. Assume that the zebra plays exactly 4 turns, and let X denote the number of distinct outcomes she registers. 2. The first moment of X is 3. The probability that both A and B are registered at some point during the 4 turns she plays is 4. The second moment of X is 2. Now assume that the zebra plays until she registers every possible outcome at least once, that is, once she has registered at least one of each of the outcomes she stops. The expected number of turns she plays is
- 50 moodle.unizwa.edu.om A coin is tossed four times. The probability of getting more than one tail is OA. 5/8 Ов. 11/16 OC. 5/16* OD. 3/8 E. None of these The probability of getting minimum three heads is Оа. 1/4 b.3/4x Ос. 5/16 Od. 3/8 OE. None of these2-137. The following circuit operates if and only if there is a pati of functional devices from left to right. The probability that each device functions is as shown. Assume that the probability that a device is functional does not depend on whether or not otiel devices are functional. What is the probability that the circureConsider randomly selecting a student at a certain university, and let V = the event that the selected individual has a Visa credit card and let M = the event that the selected individual has a Mastercard. Suppose that P(V) = 0.65, P(M) = 0.43, and P(V and M) = 0.25 a. Find the probability that the selected individual has a Mastercard or Visa. b. Find the probability that the selected individual does not have a Visa.