All parts please 2. You have two boxes. Each box contains numbers. Box A contain
ID: 3324153 • Letter: A
Question
All parts please 2. You have two boxes. Each box contains numbers. Box A contains six numbers: 1,2, 3, 4, 5, and 6 Box B contains three numbers: 1, 2, and 3. (a) Draw one number at random from each box. Assume draws are independent i. (5pts) What is the probability of drawing at least one 37 ii. (5pts) Given that the sum of the two numbers is 7, what is the probability that you drew at least one 3? ii. (5pts) What is the probability that the number from Box A is greater than b) Draw repeatedly from Box B n times with replacement, where n is large. the number from Box B? For the following problems, you can leave the probability expressions unsim- plified. i. (5pts) What is the probability that you draw exactly four 2s in the first six draws? ii. (5pts) What is the probability that the first 2 occurs on the tenth draw? iii. (5pts) True or false: The average of the n numbers should be close to 2.5. If true, state the relevant theorem(s). If false, describe why.
Explanation / Answer
Box A : 1,2,3,4,5,6
Box B : 1,2,3
Probability of drawing any random number from Box A is = 1/6
Probability of drawing any random number from Box B is = 1/3
i) P(drawing at least one 3) = P(drawing a 3 from Box A and drawing any number except 3 from Box B)+P(drawing a 3 from Box B and drawing any number except 3 from Box A)+P(drawing two 3's from the two Boxes)
= (1/6)*(1-1/3)+(1/3)*(1-1/6)+(1/6)*(1/3)=(1/9)+(5/18)+(1/18)=4/9=0.4444
ii) P(drawing at least one 3 given the sum of the two numbers is 7)
=P(drawing a 3 from Box B and drawing a 4 from Box A) [this is the only possibility]
=(1/3)*(1/6)=1/18=0.05556
iii) P(the number from Box A is greater than the number from Box B)
=P(2 from Box A) P(1 from Box B)+P(3 from Box A)[P(2 from Box B)+P(1 from Box B)]+P(4 from Box A)[P(3 from Box B)+P(2 from Box B)+P(1 from Box B)]+P(5 from Box A)[P(3 from Box B)+P(2 from Box B)+P(1 from Box B)]+P(6 from Box A)[P(3 from Box B)+P(2 from Box B)+P(1 from Box B)]
=(1/6)*(1/3)+(1/6)*(2/3)+(1/6)*1+(1/6)*1+(1/6)*1=1/18 + 1/9 + 1/2 = 1/18 + 11/18 = 2/3 = 0.667