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Statistics: The Multiplication Rule Gently Explained

Multiplication is a surprisingly powerful tool in statistics. It enables us to solve a vast amount of problems with relative ease. One thing to remember though is that the multiplication rule, to which I’ll get in a bit, only works for independent events. So let’s talk about those first.

When we roll a dice, there’s a certain probability that the number six will show. This probability does not depend on what number we rolled before. The events “rolling a three” and “rolling a six” are independent in the sense, that the occurrence of the one event does not affect the probability for the other.

Let’s look at a card deck. We draw a card and note it. Afterward, we put it back in the deck and mix the cards. Then we draw another one. Does the event “draw an ace” in the first try affect the event “draw a king” in the second try? It does not, because we put the ace back in the deck and mixed the cards. We basically reset our experiment. In such a case, the events “draw an ace” and “draw a king” are independent.

But what if we don’t put the first card back in the deck? Well, when we take the ace out of the deck, the chance of drawing a king will increase from 4 / 52 (4 kings out of 52 cards) to 4 / 51 (4 kings out of 51 cards). If we don’t do the reset, the events “draw an ace” and “draw a king” are in fact dependent. The occurrence of one changes the probability for the other.

With this in mind, we can turn to our powerful tool called multiplication rule. We start with two independent events, A and B. The probabilities for their occurrence are respectively p(A) and p(B). The multiplication rule states that the probability of both events occurring is simply the product of the probabilities p(A) and p(B). In mathematical terms:

p(A and B) = p(A) · p(B).

A quick look at the dice will make this clear. Let’s take both A and B to be the event “rolling a six”. Obviously they are independent, rolling a six on one try will not change the probability of rolling a six in the following try. So we are allowed to use the multiplication rule here. The probability of rolling a six is 1/6, so p(A) = p(B) = 1/6. Using the multiplication rule, we can calculate the chance of rolling two six in a row: p(A and B) = 1/6 · 1/6 = 1/36. Note that if we took A to be “rolling a six” and B to be “rolling a three”, we would arrive at the same result. The chance of rolling two six in a row is the same as rolling a six and then a three.

 Can we also use this on the deck of cards, even if we don’t reset the experiment? Indeed we can. But we have to take into account that the probabilities change as we go along. In more abstract terms, instead of looking at the general events “draw an ace” and “draw a king”, we need to look at the events A = “draw an ace in the first try” and B = “draw a king with one ace missing”. With the order of the events clearly set, there’s no chance of them interfering. The occurrence of both events, first drawing an ace and then drawing a king with the ace missing, has the probability: p(A and B) = p(A) · p(B) = 4/52 · 4/51 = 16/2652 or 1 in about 165 or 0.6 %.

For examples on how to apply the multiplication rule check out Multiple Choice Tests and Monkeys on Typewriters.

Credit Cards For College Students: Finding The Best Available

Student credit cards are geared primarily toward college students. But there are many factors that can make credit cards for college students the right choice for young people. So, it is very important for all consumers, not just students, to first learn about each type of available card and then choose the one that is most suitable.

Secured credit cards are one type of card for students to consider. These cards are funded in advance of purchases and do not actually extend a line of credit in the form of a loan. Rather, the cardholder sends money to the card ahead of time and uses those funds to make purchases later. In essence, a secured credit card is a bank account that does not earn interest, but can be accessed easily with the swipe of a credit card.

Secured credit cards for college students are a popular choice with many students and their parents. One of the reasons for the popularity of these student credit cards is the fact that it is not necessary to have a credit history in order to receive the card. Of course, most college students have not yet had the opportunity to build a credit history. Therefore, a secured credit card is an attractive option. In addition, secured student credit cards typically offer instant approval and do not require employment verification or even a bank account in order to receive a card.

Secured credit cards are also popular with parents because they can “load” the credit cards with as much money as they see fit for their college student. Loading a credit card is simply placing money on the card. Parents can generally choose to have money directly added to the card with each paycheck. Or, they can send money through the mail in the form of a money order or cashier’s check. There usually are also banks that will accept payments to be added to the credit card.

With a secured student credit card, parents can essentially provide their college-going child with an allowance to pay for food, school materials, or any other need the student may have. At the same time, there is no risk of the college student building a huge debt on an unsecured credit card. Once the money is spent, there is no more for the college student to spend. Secured credit cards for college students are a great way for parents to help teach their children to be responsible and independent while still providing a little help along the way.

Another benefit to using secured credit cards for college students is that many report to the major credit bureaus. In this way, the college student can begin building credit without the concern of harming his or her credit rating by being unable to pay the debt off.

For some college students, secured credit cards are not the most attractive option. One reason is because there tends to be a great number of fees associated with secured credit card. Theses fees include application fees, processing fees, and annual fees. There is generally also a fee associated with loading funds onto the credit card. Though these fees usually range from $1 to $5, the fees can add up over time.

Another reason secured credit cards may not be attractive to a college student is because the student is truly on his or her own and unable to receive financial assistance from the parents. Or, the college student may simply not have the funds available to place on a secured credit card ahead of time.

No matter the reason, unsecured student credit cards are also a popular option with credit cards. Credit cards geared toward college students are specifically designed for individuals with little credit history. Often, the Annual Percentage Rate (APR) on these cards is higher than average. Therefore, it is best for the college student to pay off the card at the end of each billing cycle whenever possible. As with secured student credit cards, unsecured credit cards for college students go a long way toward building the student’s credit history.

Bankruptcy And Students: Many Students Fail To Pay Off Their Debt

Young people in their early twenties, of which many are students, are becoming a fast-growing number of bankruptcy filers. Bankruptcy and students seems to be becoming a problem, and according to recent surveys, it is believed that teenagers younger than nineteen years of age own at least one credit card of their own. Also, it is reported that two thirds of undergraduate students have a minimum of one open credit card account, and it is believed that the average student graduates owes three to four thousand dollars in credit card debt along with other debts.

Managing Student Finances for the First Time May be a Reason for Defaulting

With more college students being marketed credit cards, it has even made some states enact legislation that limits solicitation to college students and recent bankruptcy reform procedures are also concerned with addressing the problem of bankruptcy and students. The reason behind bankruptcy and students becoming a big problem could lie in the fact that college students are learning to live alone and manage their own money for the first time, and thus find it hard to keep track of their credit card purchases.

According to experts, people tend to shop more with credit cards than when spending cash. When interest, late charges, increase in minimum payments are factored in, it makes for difficulty in managing finances and thus leads to bankruptcy and students becoming a growing malpractice.

Bankruptcy and students loans that are not repaid can often make a student feel as if he or she has just graduated from the school of hard knocks. Bankruptcy is not the escape route that students may be thinking of taking in order to avoid paying back government backed student loans as well as school loans backed by non-profit agencies. These loans are not discharged in a bankruptcy and have to be paid back after bankruptcy, though if a student can prove (very difficult actually) that the loan constitutes a considerable hardship, it can be got rid off without repayment.

Student loans, under normal circumstances, cannot be discharged under any chapter of the Bankruptcy Code. By using loopholes in government legislation, bankruptcy seems to offer an escape route to avoid paying off student loans, and the number of students that used bankruptcy to avoid paying off their debts increased dramatically over the recent past few years.

The bottom line is that it is the bankruptcy judge that has the final say, and for the lucky student, the odd bankruptcy judge may allow him or her to discharge the loan by filing for bankruptcy. Lenders too, cannot send their bills to a student who is in bankruptcy and need to wait till the case is decided. Often, it is better for the student to deal directly with the lender and find a mutually agreeable way of settling the debt, rather than going in for bankruptcy to avoid repayment.