#if p=q=0.5 then the difference is gone

[amber-tortoiseshell]

Hello! I just saw your cat colors master post and you seem very knowledgeable on the subject so I have a question. I was reading about the O/o gene for a little school assignment to talk about X linked genes. And since male cats only need one X to show orange color or black color. Wouldn't there be more orange and black male cats than female cats? I always hear people say an orange cat is more likely to be male than female but I never hear people say a black cat is more likely to be male than female. Is there some way for cats to be black that isn't linked to the O gene that changes the ratio of female to male black cats? Or are people just not noticing the ratio in black cats vs orange cats for some reason?

Yes, you're right, there are more black males than females, because the tortoiseshells are diminishing the numbers of both the black and and the red females.

Let's calculate the ratios; let p be the frequency of the black and q is the red allele. (This is also their probability: if you choose a random allele from the cat population, it'll be black with the chance of p and red with the chance of q.)

Black male: p/2 (1/2 of all cats are males and with p chance a male cat has the black allele therefore he is black)

Red male: q/2 (similarly)

Black female: p²/2 (both of her alleles need to be black, so we need to choose twice)

Red female: q²/2 (similarly)

Tortoiseshell female: pq (we need to choose a black and a red allele, and we can do it in two orders, so this is actually the more simple form of 2pq/2)

Let's check: p/2 + q/2 + p²/2 + q²/2 + pq = (p+q+p²+q²+2pq)/2 = 1, because p+q=1 and p²+2pq+q²=(p+q)²=1²=1. Good.

What could be the frequency of the red allele? I don't know, but for the sake of simplicity let's say it 0.2 (so every fifth male cat is orange).

Black male: p/2=0.8/2=0.4 (40%)

Red male: q/2=0.2/2=0.1 (10%)

Black female: p²/2=0.8²/2=0.64/2=0.32 (32%)

Red female: q²/2=0.2²/2=0.04/2=0.02 (2%)

Tortoiseshell female: pq=0.8*0.2=0.16 (16%)

So this means the male:female ratio in red cats is 10:2=5:1 (for every five males there's one female), while in blacks 40:32=5:4 (for every five male there're four females).

You see why it's much easier to notice.

[lawsofemotion]

Basic Connectives and Truth Tables

The summation formula was obtained by counting the same collection of objects in two different ways and then evaluating the results. This formula was established by a combinatorial proof. This is one of the many different techniques for arriving at a proof. Chapter 2 discusses what constitutes a valid argument and what constitutes a more conventional proof. When a mathematician wishes to provide a proof for a given situation, they must use a system of logic. The logic of mathematics is applied to decide whether one statement follows from, or is a logical consequence of, one or more other statements.

Statements and Compound Statements In the development of any mathematical theory, assertions are made in the form of sentences. These verbal or written assertions are called statements or propositions, and they are declarative sentences that are either true or false, never both. Lowercase letters, such as p, q, and r, are used to represent these statements.

The following are examples of statements, where they could be either true or false, but not both:

p: Combinatorics is a required course for sophomores. q: Margaret Mitchell wrote Gone with the Wind. r: 2 + 3 = 5

However, exclamation sentences, such as "What a beautiful evening," and command sentences, such as "Get up and do your exercises," are not regarded as statements, because these sentences can be either true or false. These sentences do not have a set truth value.

The above statements labelled p, q, and r are called primitive statements, since there is no way to make the statement any more simplified.

Example The sentence, "The number x is an integer," is not a statement, because its truth value (true or false) cannot be determined until a numerical value is assigned for x. If x was assigned the value of 7, the result would be a true statement. However, if x was assigned 0.5, √2, or π, the result would be a false statement.

Statements can create new statements in two ways: negation and compounding.

1. Negation. Transform a given statement p into its negation statement ¬p, read as, "not p."

For example, the above statement p: "Combinatorics is a required course for sophomores." has the negation statement ¬p: "Combinatorics is not a required course for sophomores."

Note that the negation of a primitive statement is not considered to be a primitive statement.

2. Compounding. There are five logical connectives to combine two or more statements to create compound statements.

2.1. Conjunction. The conjunction of the statements p, q is denoted as p ᴧ q, which is read as, "p and q."

For example, the compound statement p ᴧ q represents the compound statement p ᴧ q: "Combinatorics is a required course for sophomores, and Margaret Mitchell wrote Gone with the Wind."

The compound statement p ᴧ q is only true if both statements p, q are true.

2.2. Disjunction. The disjunction of the statements p, q is denoted as p ᴠ q, which is read as, "p or q."

For example, the compound statement p ᴠ q represents the compound statement p ᴠ q: "Combinatorics is a required course for sophomores, or Margaret Mitchell wrote Gone with the Wind."

The compound statement p ᴠ q is only true if the statement p is true, or the statement q is true, or both the statements p, q are true. The compound statement p ᴠ q uses the English term "and/or."

2.3. Exclusive "or". The exclusive "or" of the statements p, q is denoted as p ⊻ q, which is read as, "p exclusive "or"/xor q."

For example, the compound statement p ⊻ q represents the compound statement p ⊻ q: "Combinatorics is a required course for sophomores, or Margaret Mitchell wrote Gone with the Wind, but not both."

The compound statement p ⊻ q is only true if either the statement p is true or the statement q is true, but they cannot both be true.

2.4. Implication. The implication of the statements p, q is denoted as p → q, which is read in five different ways: i. "p implies q", ii. "if p, then q", iii. "p is sufficient (enough) for q", iv. "q is a necessary condition for p", v. "p only if q".

For example, the compound statement p → q represents the compound statement p → q: "If combinatorics is a required course for sophomores, then Margaret Mitchell wrote Gone with the Wind."

The statement p in p → q is called the hypothesis, and the statement q in p → q is called the conclusion. When statements are compounded using implication, the statements do not need to have a casual relationship with each other for the compound statement to be true.

2.5. Biconditional. The biconditional of the statements p, q is denoted as p ↔ q, which is read as, "p if and only if (iff) q."

For example, the compound statement p ↔ q represents the compound statement p ↔ q: "Combinatorics is a required course for sophomores if and only if Margaret Mitchell wrote Gone with the Wind."

Truth Tables The above discussed the circumstances for when the statements p ᴠ q, p ⊻ q are considered true, using the basis of the truth values of their components p, q. The following table demonstrates a truth table of the negation of p based on the truth value of the component p, where "0" represents false and "1" represents true.

Note that the truth values for p and ¬p are opposite.

The following table demonstrates the truth table of the compound connectives based on the truth values of their components p, q:

Note that, since there are two primitive statements p, q, there will be a total of four possible truth value assignment cases. In general, if there are n primitive statements, there will be a total of 2n possible truth values assignment cases for each statement.

Note that for p ᴧ q to be true, both the statements p and q have to be true. For p ᴧ q to be false, both the statements p and q have to be false or either one has to be false.

Note that p → q is true if the statement p is false and the statement q is true. This is because the statement q can be true even if the statement p is false. For example, consider the statement p → q: "If I go to the movies, then the sky is blue." If the conclusion q is true, then it does not matter if the hypothesis p is true, because it will be a blue sky no matter if they go to the movies or not. The only time the statement p → q is false is if the statement p is true yet the statement q is false. This does not make sense for it to be true, because the statement q is a necessary condition for the statement p, and so if the statement q is false and yet the statement p is true, then the statement p → q cannot be true. It does not make sense for an implication to be true if a true statement leads to a belief that is actually false. However, even if both the statements p and q are false, p → q is considered true.

Note that the biconditional truth values are true if both the statements p and q are true or both the statements p and q are false.

Example Let s, t, and u denote the following primitive statements: s: Phyllis goes out for a walk. t: The moon is out. u: It is snowing. These three primitive statements can create the following compound statements written symbolically: 1. (t ᴧ ¬u) → s: If the moon is out and it is not snowing, then Phyllis goes out for a walk. 2. t → (¬u → s): If the moon is out, then if it is not snowing, Phyllis goes out for a walk. 3. ¬(s ↔ (u ᴠ t)): It is not the case (¬) that Phyllis goes out for a walk (s) if and only if (↔) the moon is out (u) or (ᴠ) it is snowing (t). Compound statements written in words can be converted to its simplified, symbolic form. 1. "Phyllis will go out walking if and only if the moon is out." The statements s, t are used with a biconditional (if and only if). The symbolic form is s ↔ t. 2. "If it is snowing and the moon is not out, then Phyllis will not go out for a walk." All three statements s, t, u are used with implication (if, then). The statements t, u use conjunction (and). The statements t, s additionally use negation (not). The symbolic form is (u ᴧ ¬t) → ¬s. 3. "It is snowing but Phyllis will still go out for a walk." Although this statement may look like implication, in logic, "but" has the same meaning as "and." Therefore, the symbolic form is u ᴧ s.

The following example will clarify better about the truth values of implication:

Example Suppose it is almost the week before Christmas and Penny will be attending several parties that week. She plans not to weigh herself until the day after Christmas. She considers what those parties may do to her weight by then, so she makes the following resolution for December 26: "If I weigh more than 120 pounds, then I shall enroll in an exercise class." Let p and q denote the resolution's primitive statements: p: I weigh more than 120 pounds. q: I shall enroll in an exercise class. The resolution is an implication statement, so her resolution can be written as p → q. Using this scenario can help understand the truth values for implication on the above truth table. In regards to row 4 of the truth table for implication, the components p and q have the truth value of 1, so they are both considered true. This means that, on December 26, Penny finds that she weighs more than 120 pounds and promptly enrolls in an exercise class, just as she said she would. Therefore, she didn't break her resolution p → q, and so p → q is considered true. In regards to row 3 of the truth table for implication, the component p has the truth value of 1 and the component q has a truth value of 0. This means that, on December 26, Penny finds her weight to be over 120 pounds, but she makes no attempt to enroll in an exercise class. Therefore, she broke her resolution p → q, and so p → q is considered false. In regards to row 1 of the truth table for implication, the components p and q both have the truth value 0. This means that, on December 26, Penny finds her weight is 120 pounds or less, and so she does not enroll in the exercise class. Therefore, although she did not follow her resolution p → q, she did not violate it either, and so p → q is still considered true. In regards to row 2 of the truth table for implication, the component p has the truth value of 0 and the component q has a truth value of 1. This means that, on December 26, Penny finds her weight is 120 pounds or less, yet she still enrolls in an exercise class. Whatever reason why she still enrolls in an exercise class, she still doesn't violate her resolution p → q, and so p → q is considered true.

Implication versus Biconditional As shown on the above truth table, implication statements and biconditional statements are not the same. This can be confusing in everyday speaking for English, because there are situations where an implication is used that could be perceived as biconditional, when it is not the case.

For example, consider the following two implications a certain parent may direct to their child:

s → t: If you do your homework, then you will get to watch the baseball game. t → s: You will get to watch the baseball game only if you do your homework.

For the implication s → t, the parent is using a positive approach by emphasizing the enjoyment in watching the baseball game. For the implication t → s, the parent is using a negative approach by emphasizing on the punishment to be incurred if the child does not do their homework.

In either case, the parent wants their implication, whether it is s → t or t → s, to be understood as biconditional s ↔ t. For the implication s → t, the parent wants to hint at the punish while promising enjoyment. For the implication t → s, the parent uses the punishment as a possible threat such that, if the child does in fact do the homework, then that child will definitely be given the opportunity to enjoy watching the baseball game.

However, in logical problems, being unambiguous is important, and so when given an implication, it should not be interpreted as a biconditional. Although definitions are a notable exception.

Example The compound statement "Margaret Mitchell wrote Gone with the Wind, and if 2 + 3 ≠ 5, then combinatorics is a required course for sophomores." can be written in the symbolic form q ᴧ (¬r → p), where the components q, r, p represent these three primitive statements above. The following is its truth table:

When constructing truth tables, the component columns are required, then if there is any negation statements, create a column for those, and then slowly build the compound statement one connective at a time. The columns of the components and the columns of the negation statements are used to determine the truth values for the columns that have connectives.

Example The following is the truth table for the compound statements p ᴠ (q ᴧ r) and (p ᴠ q) ᴧ r:

Note that parentheses are used if there is more than two statements in the compound statement. This makes compound statements readable and truth tables easier to construct.

Example The following is the truth table for the compound statements p → (p ᴠ q) and p ᴧ (¬p ᴧ q):

Notice how the column of the compound statement p → (p ᴠ q) has all truth values of 1, and the column of the compound statement p ᴧ (¬p ᴧ q) has all truth values of 0. When this happens, the two compound statements are considered a special type of statement.

Tautologies and Contradictions If a compound statement is true for all truth value assignments from its component statements, the compound statement is called a tautology, denoted as T0. If a compound statement is false for all truth value assignments from its components, the compound statement is called a contradiction, denoted as F0.

Tautologies and implications can be used to describe a valid argument. In general, an argument starts with a list of give statements called premises which form the hypothesis, and what follows is another statement called the conclusion of the argument.

The premises p1, p2, ..., pn are examined to show that the conclusion q follows logically from these given premises, trying to show that if each premise p1, p2, ..., pn is a true statement, then the statement q is also true. Tautologies, or valid arguments, are of the following form:

(p1 ᴧ p2 ᴧ ... ᴧ pn) → q

The conjunction statement p1 ᴧ p2 ᴧ ... ᴧ pn of n statements is true if and only if pi, 1 ≤ i ≤ n is true.

Note that the conclusion q does not have to be a primitive statement.

The hypothesis is the conjunction of the n premises. If any one of the n premises is false, the whole conjunction statement is false, and so it does not matter what the truth value is for the conclusion q, because the implication is always true if the hypothesis is false. If all of the n premises are true, and under these circumstances the conclusion q happens to be true too, then the implication is also true, and therefore is a tautology and a valid argument.

PDF reference: 75

For a), the sentence is a primitive statement, because it can be either true or false.

For b), the sentence is not a statement, because it is not known whether the sentence is true or false.

For c), the sentence is a primitive statement, because it can be either true or false.

For d), the sentence is an implication (if, then) statement and so therefore a statement.

For e), the question is not a statement, because it is a question.

For f), the sentence is a primitive statement, because it can be either true or false.

The only way for an implication to be false is the hypothesis to be true and the conclusion to be false. Therefore, for each question, it will only look at one specific row, which is the row that has the statement p as false and the statement q as true.

a)

b)

c)

d)

For a), the statement uses implication, and so the symbolic form is r → q.

For b), the statement uses implication, and so the symbolic form is q → p.

For c), the statement uses conjunction and implication, and so the symbolic form is (s ᴧ r) → q.

For a), q → p: If the triangle ABC is equilateral, then it is isosceles.

For b), ¬p → ¬q: If the triangle ABC is not isosceles, then it is not equilateral.

For d), p ᴧ ¬q: The triangle ABC is isosceles, but it is not equilateral.

For a), "If Darci practices her serve everyday, then she will have a good chance of winning the tennis tournament."

For b), "If you do not fix my air conditioner, then I will not pay the rent."

For c), "If Mary is allowed on Larry's motorcycle, then she must wear a helmet."

Note that, for c), it uses "p only if q."

a)

e)

g)

For a), right now m = 3, n = 8. The if-statement executes, because n - m = 5 is true. Therefore, n = n - 2 = 8 - 2 = 6.

For b), now m = 3, n = 6. The if-statement executes, because both the statements 2 * m = n and [n/4] = 1 are true. [n/4] = 1 are true, because 6/4 = 1.5, but the program will drop the decimal. Therefore, n = 4 * m - 3 = 4 * 3 - 3 = 9.

For c), now m = 3, n = 9. The if-statement does not execute, because neither n < 8 nor [m/2] = 2 are true. Therefore, the else-statement executes, and so m = 2 * n = 2 * 9 = 18.

For d), now m = 18, n = 9. The if-statement executes, because both the statements m < 20 and [n/6] = 1 are true. Therefore, m = m - n - 5 = 18 - 9 - 5 = 4.

For e), now m = 4, n = 9. The if-statement does not execute, because neither n = 2 * m nor [n/2] = 5 are true. Therefore, since there is no else-statement, m = 4 and n = 9.

[cryptobully-blog]

Markets Now: Dow Gets Whipsawed by Fed

http://cryptobully.com/markets-now-dow-gets-whipsawed-by-fed/

Markets Now: Dow Gets Whipsawed by Fed

The Dow Jones Industrial Average was sitting pretty at the end of January. And then it wasn’t, as a correction hit. Since then it’s gone up and down and up again, and the wild swings don’t seem likely to end any time soon. With that in mind, we’re keeping a semi-live look on the volatile markets. Here’s the latest from Barron’s reporters…

3:26 p.m. I hate using the word “whipsaw” to describe the market action, but there really is no better one today. The Dow was up. Then it was down. Now it’s up 72.34 points, or 0.3%, at 24,799.61. See, whipsawed. And I wouldn’t be surprised if the blue-chip benchmark finished up 200 points or in the red.

2:53 p.m. Wow. The Dow’s gone from 200 points up to down on the day in almost no time at all. Its fallen 14.17 points, or 0.1%, to 24,713.10, while the S&P 500 has, declined 0.2% to2,712.55 and the Nasdaq Composite has dropped 0.4% to 7,335.62. Could the Fed’s inflation expectation be the cause? “The most notable change to the economic forecasts was the shift higher in the inflation expectations, which show the central tendency above the 2.0% target in 2020,” writes Jefferies Senior Money Market Economist Thomas Simons. “This is something that we have not previously seen from the Fed’s forecast.”

2:07 p.m. Stocks are rising after the Federal Reserve lifted its benchmark interest rate by a quarter point but left its forecast for two more hikes this year unchanged.

The S&P 500 has risen 0.5% to 2,729.46, while the Dow Jones Industrial Average has gained 204.21 points, or 0.8%, to 24,931.48. The Nasdaq Composite has advanced 0.4% to 7395.29. The 10-year Treasury yield has ticked up 0.035 percentage point to 2.91%, while the US Dollar Index has slipped 0.3% to $90.13.

“Pretty much everyone saw this coming, but it’s still a significant event,” writes E*Trade’s Mike Loewengart. “We are another confident step closer to a normalized rate environment.”

Now we just need to see what Fed Chairman Jerome Powell says in his first Q&A.

12:12 p.m. Well, that was quite an hour. When I published my last update, nothing was happening. Then the Dow Jones Industrial Average decided to take off, at least a little bit. The Dow has gained 123.54 points, or 0.5%, to 24,850.81, while the S&P 500 has risen 0.4%, to 2728.02, and the Nasdaq Composite has advanced 0.3%, to 7386.09.

Maybe the market’s betting on a dovish surprise?

10:54 a.m. The stock market may not be doing much ahead of today’s Federal Reserve meeting, but that doesn’t mean there isn’t at least a little bit of action elsewhere. Sure, the S&P 500 has ticked up 0.1% to 2719.05, while the Dow Jones Industrial Average has dipped 10.76 points to 24,716.51, and the Nasdaq Composite has advanced 0.1% to 7373.21. That’s certainly nothing to write home about. But the U.S. Dollar Index has dropped 0.3% to 90.09 today, and we’ll be keeping a close eye on whether the currency leads to more dollar weakness or a rebound.

And then there’s the Treasury market. So far today, the 10-year Treasury yield has risen 0.025 percentage point to 2.901% and the 2-year Treasury yield has risen to 2.337%. The difference of 0.564 percentage point between the two is greater than yesterday’s close of 0.545 point–good news for those worried about the flattening yield curve. “If the net result of today’s FOMC meeting is that the 10’s -2’s spread drops to fresh lows (so below 0.50%) I’ll take that as a pretty significant caution signal on markets,” writes Sevens Report’s Tom Essaye.

6:57 a.m. It’s all about the Federal Reserve today, so it shouldn’t come as a surprise that the market isn’t doing much this morning. S&P 500 futures have ticked up half a point, while Dow Jones Industrial Average futures have risen 5 points. Neither is enough to move the indexes by more than a fraction of a percentage point. Nasdaq Composite futures have fallen 0.3%, but probably would be right around flat if it weren’t for the Facebook (FB) saga.

So what’s the Fed going to do? We know one rate hike is priced into the market, so anything beyond that would be a shock, and few appear to expect that. Instead, all eyes will be on the Feds “dots,” the individual forecasts that can be extrapolated into a prediction for the number of rate hikes to come. Right now, the dots suggest three hikes in 2018 and two in 2019, and the market will be watching to see if those change, says Jones’ Trading’s Michael O’Rourke. “From our perspective, there has been just enough weakness and there are still enough doves at the Fed to keep the forecast at 3 hikes, although many in the market are expecting the upward shift to four,” he says.

Now we just have to wait to see how the market responds.

Markets