CS 237 Fall, 2017, Homework 01 Solution

Due Thursday, 9/14, at Midnight through Websubmit

For the following problems, analyze means to specify (i) the sample space S, (ii) probability function P, (iii) the events listed (i.e.,list the members of each event), and (iv) their probabilities. In some cases, additional information may be required. You may abbreviate or schematize as necessary, as long as the answer is perfectly clear.

Sometimes it is useful to first write down a "pre-sample space" which helps to think about the actual sample space. This is often useful when the literal outcome of the random experiment is non-numeric but the sample space is numeric.

Example: Toss a fair coin and report the number of heads that appear. Let A = "one head appears." 
Analyze, providing the "pre-sample space" (the appearance of the two sides of the coin).

Solution:  The pre-sample space is { T, H }. Then:
       
           S = { 0,   1   }
           P = { 0.5, 0.5 }
           A = { 1 }
           P(A) = 0.5

Quantitative answers may be given in fractional form, as long as they are reduced (e.g., 1/5 instead of 20/100), or as decimals; in the latter case round to 4 decimal places, with trailing zeros suppressed (i.e., 0.2500 would be expressed as 0.25). If rounding to 4 places would give 0.0000, then give the result in scientific notation with 4 significant digits with trailing digits suppressed, e.g., 0.00000023 would be expressed as 2.3 * 10-7.

1. Suppose that a study is being done on all families with 1, 2, or 3 children (all having different ages, i.e., no twins), and let the outcomes be the genders (G = girl and B = boy) of the children in each family in ascending order of their ages (e.g., BG means an older girl and a younger boy). Assume all possible configurations of genders and numbers of children is equally likely (i.e., this will be an equiprobable probability space). Let events A = "families where the oldest child is a boy" and B = "families with exactly two girls and any number of boys." Analyze. (No pre-sample space necessary.)

2. Suppose that each time Wayne charges an item to his credit card, he rounds the amount to the nearest dollar in his records (assume that for x dollars, the amount x.50 is rounded to x + 1 dollars). The round-off error is defined as (recorded - actual); the units are dollars, so if Wayne charges $4.25, he records it as $4, and the round-off error is $-0.25, but if he charges $4.75, the value recorded is $5 and the round-off error is $0.25. Assume this is random, so that each time Wayne charges to his card, he performs a equiprobable random experiment whose outcome is the round-off error. Let event A = "at most 3 cents is rounded off in either direction" (i.e., | recorded - actual | ≤ 0.03). Analyze. (No pre-sample space necessary.)

3. Suppose an experiment consists of two steps. First a coin is tossed; if the result is heads, then a six-sided die is thrown; if the result of the coin toss is tails, then the coin is tossed again (no die involved). Let A = "at least one head is thrown during the experiment" and B = "an odd outcome in the case the die is tossed". Analyze. This is not equiprobable, so a tree diagram may help. You do not need to give the tree diagram in your answer

4. Consider the random experiment of flipping a fair coin until a head appears. The result of the random experiment is the number of flips. Let A = "it takes an odd number of flips." Give the probability P(A). (No need to analyze, since we did it in class.) Show your reasoning, not just give the answer. (Hint: compare the sequence of probabilities in the case of an odd number of flips, and the sequence of probabilities in the case of an even number.)

5. Consider the random experiment of flipping a fair coin until the second head appears. (The heads need not occur one after the other.) The result of the random experiment is the number of flips. Analyze. Hint: think of it as repeating the previous experiment twice, and summing the total number of flips. Draw a 2D grid, similar to the one we used for tossing two dice in lecture 9/12, but infinite:

       
                   1  2  3  4  .....
                 1
                 2
                 3
                 .

Since the two trials are independent, you can figure out the probability of each square, and then sum across the diagonals as in the dice example. Prove that this is indeed a probability space by showing that the sum of the probabilities is indeed 1.0.

6. Two students are supposed to take the final exam in CS 237 but don't show up. They come to Snyder's office several hours after the exam and explain that they had to drive to visit a sick friend the morning of the exam, but got a flat tire on the way back, and missed the exam; they ask to be given a makeup exam. Snyder, with a mysterious smile, agrees. When they get the exam, it consists of two problems, the first (worth 5%) a simple problem from the course, and the second (worth 95%) consists of the single question "Which tire was it?" If the students have a 10% probability of actually having had a flat tire, then what is their chance of passing the exam? Assume that if they didn't have a flat tire, they have to guess randomly at the exam, and if they did indeed have a flat time, they would both remember which one. Assume they drove a car and the car has 4 tires. Let A = "they pass the exam." Analyze, giving a tree diagraming the random experiment (do the best you can to format it, don't worry about making it pretty).

Consider an ordinary deck of cards:

7. Supposing you shuffle the deck thoroughly and draw a single card, give the probability that the card is:

(a) the King of Diamonds
(b) a black card
(c) not a face card (i.e., not Jack, Queen, King)
(d) a spade or an Ace (hint: remember that or in English is the "inclusive or" not the "exclusive or")

8. Suppose we throw a dart at a square target 1 meter on a side, which has a bullseye in the center of radius 0.1 meters. Assume the dart lands with equal probability anywhere inside the square target. Give the probabilities of the following events:

(a) The dart lands in the bullseye;
(b) The dart lands within 0.1 meter of an edge of the square target (i.e., the shortest distance from the dart to the closest edge is ≤ 0.1);
(c) The dart lands within 0.1 meter of a corner of the square target (i.e., the distance from the dart to the closest corner point is ≤ 0.1);
(d) The dart lands in the exact center of the square (equivalently, in the exact center of the bullseye).

9. Suppose we consider the random experiment of randomly choosing a real number x in the range [0..1) for example using a spinner as discussed in class. Give the probability of the following events occurring.

(a) x is larger than 0.5 but smaller than 0.61 (probability problems are often written, unfortunately, in English, so you have to translate, in this case into "0.5 < x < 0.61")

(b) x is larger than or equal to 0.5 but smaller than 0.61 (i.e., 0.5 ≤ x < 0.61)

(c) x is one of the values 0.1, 0.3, 0.5, or 0.545

(d) x is one of the values in the infinite list 0.1, 0.11, 0.111, 0.1111, etc. (a finite sequence of 1s)

(e) x is a rational number (can be expressed as a fraction)

Hint: Consider the role of Axiom 3 in (c), (d), and (e).

10. Consider the formula: P( ( A ∩ Bc ) ∪ ( Ac ∩ B ) ) = P(A) + P(B) - 2*P(A ∩ B). (This is the "symmetric difference" that I mentioned at the very first lecture.)

(a) Draw a Venn Diagram illustrating this formula and explain in words why it is true.

(b) Prove this formula using the axiomatic method described in lecture on Tuesday 9/12 (you may use any of the formulae that I proved in class as lemmas in your proof if you wish--you do not need to restate the proofs I gave).

 

[Lab Problems: You will also be asked to submit your lab problems with this homework. These will be presented at lab on Monday.]