Homework 2

STAT 225—Homework 2, Fall 2023 — DUE at the beginning of class on Fri. September 22, 2023 If you are ill on the due date or have been ordered to isolate/quarantine—email stat225questions@stat.purdue.edu IMMEDIATELY! You will need verification of this. You will write out your homework on paper or print it from a word document or pdf. (DO NOT rewrite the problem itself.) Clearly label the parts of the problem and show your work. Write your name at the top of EACH page. You must SHOW WORK to receive full credit!! When finding a probability, you MUST write out a probability statement using proper notation!! Use 4 decimal places when rounding your final answers. 1. Two independent random variables (r.v.’s), 𝑋𝑋 and 𝑌𝑌 , have the following properties.  𝑋𝑋 is always negative  𝑌𝑌 is always positive 1  𝐸𝐸 [ 𝑋𝑋 2 ] = 23  𝑉𝑉 𝑉𝑉𝑉𝑉 [ 𝑋𝑋 ] = 7  𝐸𝐸 [ 𝑌𝑌 ] = 6  𝑉𝑉 𝑉𝑉𝑉𝑉 [ 𝑌𝑌 ] = 3 Let 𝑊𝑊 = 2 𝑋𝑋 + 𝑌𝑌 and 𝑉𝑉 = 𝑋𝑋 − a) 𝐸𝐸 [ 𝑌𝑌 2 ] b) 𝐸𝐸 [ 𝑋𝑋 ] c) 𝐸𝐸 [ 𝑊𝑊 ] d) 𝐸𝐸 [ 𝑉𝑉 ] 𝑌𝑌 , use properties of expected value and variance to find the following. 2 e) 𝑉𝑉 𝑉𝑉𝑉𝑉 ( 𝑉𝑉 ) f) 𝑉𝑉 𝑉𝑉𝑉𝑉 ( 𝑊𝑊 ) g) 𝐸𝐸 [2 𝑊𝑊 − 𝑉𝑉 + 4] 2. A local restaurant is currently offering a carry-out dinner special. The customer is able to pick one item from each of the lists below.  Main dish: chicken, pork, beef, or tofu  Starch: potatoes, corn, rolls, rice, or pasta  Vegetable: broccoli or spinach  Dessert: pie, cake, or ice cream a) How many different meals can a customer order? 4*5*2*3 =120 b) What is the probability that a customer would place an order that contains chicken and spinach? P(Chicken and Spinach ) = 0.25*0.5 = 0.125 c) What is the probability that a customer will not order corn? P(no Corn) = 4/5 = 0.8 d) Given that a customer orders the vegetarian main dish, what is the probability that they also order cake for dessert? P(cake | tofu) = P(Cake and Tofu) / P(tofu) = (10/120 )/ 0.25 0.333 e) The restaurant has sold out of pork and broccoli. How many different meals can be created now? 3*5*1*2 = 30

3. The Super Secret Gaming Club likes to make up passwords using by arranging unusual symbols. This week they are using the following 10 symbols: ©©©© Ψ Ψ ∞∞∞ ♡ in any order. There are a total of 10 characters in the code. a) How many different codes are possible? 10! / (4!2!3!) b) What is the probability that a randomly chosen code, the first and fourth symbol are © ? 4/10 * 3/9 c) What is the probability that a randomly chosen code ends with either Ψ or ♡ ? 3/10 d) What is the probability that in a randomly chosen code, the three ∞ symbols are next to each other? e) 10!/ (4!2!1!) / total f) What is the probability that in a randomly chosen code, the two Ψ symbols are NOT next to each other? 1 – p( Pitchfork next to eachother) = 1 - ( 10! / (4!1!3!)) / total g) What is the probability that in a randomly chosen code, the first and the last symbols are the same? (4/10 * 3/9) + (2/10*1/9) + (3/10 * 2/9)

h) How many bottles of hand soap does a household use per 10-years on average?

6. You visited Perry’s Pumpkin Patch and bought a box full of mini pumpkins for decorations. In the box are 11 orange pumpkins, 6 green pumpkins, and 3 white pumpkins. The size and shape of the pumpkins are similar. You reach into the box (without looking) and pull out the pumpkins one at a time as described in each scenario below. a) Select 2 pumpkins with replacement. Find the probability the second pumpkin is white. P(2 nd is white) = 3/20 = 0.15 b) Select 2 pumpkins with replacement. Find 𝑃𝑃 ( 2 𝑛𝑛𝑛𝑛 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝 𝑛𝑛 𝑝𝑝 𝑖𝑖 𝑤𝑤 ℎ 𝑝𝑝 𝑖𝑖 𝑖𝑖 | 1 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝 𝑛𝑛 𝑝𝑝 𝑖𝑖 𝑤𝑤 ℎ 𝑝𝑝 𝑖𝑖 𝑖𝑖 ) . P( 2 nd is white | first is white ) = P(first and second is white) / P (first is white) = 0.0225 / 0.15 = 0.15 c) With replacement, is the color of 1 st pumpkin independent of the color of 2 nd pumpkin? Mathematically support your answer. P(White | White) = 0.15. P(white) = 3/20 = 0.15 Equal is so events independent d) Select 2 pumpkins without replacement. Find the probability that the second pumpkin is green. P(second is green) = 6/20’*5/19 + 14/20*6/19 = 0.078947 +0.22105 = 0.3 e) Select 2 pumpkins w i th o u t replacement. Find 𝑃𝑃 ( 2nd pumpkin is gre e n | 1 st pumpkin is gre e n ) . P(second is green | first is green) = p(First and second is green) / P f(irst is Green) = 0.07894 / 0.3 = 0.263157 f) Without replacement, is the color of 1 st pumpkin independent of the color of 2 nd pumpkin? Compare it with your answer in c). P(second is green | first is green) = 0.263157 ; P(second is green) = 0.3; Not equal so not independent 7. In a poker game, 5 cards are dealt from a standard 52 card deck that has been well shuffled. You are the only player in this scenario. (Note: if you are not familiar with poker hands, you may want to look up what some of these are.) a) How many different 5-card hands are possible? 52C5 = 2598960 b) What is the probability that you are dealt two pairs? Choose two suits 4C2 Choose two numbers 13C2 Choose last card number and suit 11C1 * 4C1 6*78*11*4 / 52C5 = 0.047539 c) What is the probability that you are dealt a 3 of a kind or 4 of a kind? (Note: 3 of a kind means that the other 2 cards are NOT a pair.) Prob of 3 of a kind 13C1*4C3*12C2*4C1*4C1 = 54912 Prob 4 of a kind 13C1*4C4 * 12C1*4C1 = 624 (54912 + 624 ) / 2598960 = 0.213685