Photo Credit: Piulet

Let’s see how you do on the following 10 questions.

1. More people die due to homicide than by suicide in the United States. True or False.

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False.

Overall, suicides greatly exceed homicides, but rates vary by age, sex, and urban or rural residence. Suicides, the 11th leading cause of death in the United States, outnumber homicides by 2 to 1. However, homicides exceed suicides among young adults. Homicide rates are highest in the most urban counties, and suicide rates are highest in the most rural counties.

Source: Health, United States, 2007

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2. Put the list in order, from highest to lowest, by the rate that they cause injury death in the United States?
• Suffocation
• Motor Vehicle Traffic
• Falls
• Firearms
• Act of War / Terrorism
• Poisoning

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1. Motor Vehicle Traffic
2. Firearms
3. Poisoning
4. Falls
5. Suffocation
6. Act of War / Terrorism

Source: CDC Injury Data

Motor vehicle traffic accidents remain the leading cause of injury death in the United States. The age-adjusted motor vehicle traffic death rate in 2004 (14.7 per 100,000 population) was 19% lower than the rate in 1985. Most of the decline occurred from 1988 through 1992, when the rate decreased an average of 4.7% per year. Since then, the rate has fluctuated, with only modest declines of less than 1% per year.

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3. Suppose you are a contestant on a game show in which you are confronted with three doors. Behind one door is a fabulous prize, while behind the other two doors there is nothing. Your task is to pick the door with the prize. The host allows you to pick a door. After you choose a door, but before you get to see what’s behind it, the host will open one of the other two remaining doors that doesn’t have a prize behind it, eliminating that door from the situation. You are then given the option to either stick with your initial choice of door, or switch to the other remaining door. What are your odds of winning the prize if you choose to stay with your current door? What are your odds of winning if you choose to switch to the other door?

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In all of statistics, no problem is more famous or counter intuitive than the Monty Hall problem. Although the statistics behind the problem are almost painfully simple, the answer is so counter intuitive that many people stubbornly insist that it can’t possibly be right, even when confronted with mathematical proof. The problem is as follows:

Most people intuitively think that the odds of winning the prize are 1/2 regardless of whether you switch; after all, there are only two doors and one of them has the prize. But in fact if you stay with your initial door you will only have a 1/3 chance of winning. By switching doors, your chances of winning the prize increase to 2/3. Thus switching doors is always the best choice.

When you choose your initial door, there is a 1/3 chance that you picked the door with the prize. That doesn’t change just because the host eliminates one of the loosing doors. Even after one door is eliminated and you have only two doors to choose, there is still only a 1/3 chance that you initially picked the correct door. Thus the odds that the other remaining door has the prize are 2/3.

Resources: Counterintuitive-Statistics and Monty Hall and Funny Math

Half way down the page on the Monty Hall and Funny Math page is a great animated video explaining the situation. If you still have questions, take a look.

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4. Suppose there is a rare disease that randomly affects 1 out of every 100,000 people. Although you know it isn’t likely that you would have the disease, you want to be sure. Fortunately there is a test available for the disease that is accurate 99.9% of the time. In other words, there is a 1 in 1000 chance that the test will falsely tell a healthy person that they have the disease, and a 1 in 1000 chance that it will falsely tell a sick person that they do not have the disease. You decide to take the test, and unfortunately the result comes back positive – indicating that you have the disease. What are the odds that the test was correct and you have the disease? What are the odds that the test was wrong and you don’t have the disease?

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Most people would intuitively assume that since the test is correct 99.9% of the time, there is a 99.9% chance that they have the disease and only a 0.1% chance that the result was a false positive. Fortunately for you, there is actually a 99% chance that the result was a false positive – even though the test is accurate 99.9% of the time.

To understand why, you must remember that there are two ways you could get a positive result on the test. The first way is for you to have the disease, which is a 1 in 100,000 chance. The other way is for the test to give a false positive result, which is a 1 in 1000 chance. Thus if the test gives a positive result, it is 100 times more likely that it is a false positive than a true positive – even though the test is 99.9% accurate.

This bit of counter intuitive statistics has very important implications for real-life laboratories that test for diseases, especially when the disease that they are testing for is rare. Such labs must always do a test multiple times to ensure that a result isn’t a false positive, even when the test being used is very accurate. If the test in the above scenario was performed twice and came back positive both times, there is still a 10% chance that you don’t really have the disease, since the odds of two false positives in a row are 1 in a million while the odds of actually having the disease are 1 in 100,000. If the test were performed three times and came back positive each time, there would be a 99.99% chance that the positive result was accurate.

Source: Counterintuitive-Statistics

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5. Suppose you run into a friend who you have not seen in many years. While discussing what’s been going on in your lives since you last met, your friend says that she has given birth to non-identical twins. You ask your friend “Is one of the twins a boy?” and she answers “Yes”. You then ask “Is the other twin a girl?” What are the odds that the person will answer yes, the other twin is a girl? What are the odds that the person will answer no, and tell you that the other twin is also a boy?

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Most people would assume that since the odds of each child being a boy or girl are 1/2, there would be a 1/2 chance that the other child would also be a boy. In fact, there is a 2/3 chance that the other child is a girl.

Once again, to understand this counter intuitive answer one must look at what possibilities could lead to each answer, and how likely each possibility is. Let’s label the two twins A and B. There are 4 equally-likely possibilities:

1. A is a boy and B is a boy (probability 1/4)

2. A is a boy and B is a girl (probability 1/4)

3. A is a girl and B is a boy (probability 1/4)

4. A is a girl and B is a girl (probability 1/4)

Since the mother has already told us that one child is a boy, we know that the 4th option isn’t correct. That leave us with possibilities 1-3, each of which are equally likely. But only one of those possibilities (number 1) results in both children being boys, while two (2 and 3) result in one child being a boy and the other a girl. Thus there is a 2/3 chance that the mother will answer “Yes, the other twin is a girl.”

Interestingly, if you change your question from “Is one of the twins a boy?” to “Was the first-born twin a boy?” the odds of the other twin being a girl drop down to 1/2. This is because asking about a specific twin (rather than both twins at the same time) allows us to eliminate two options rather than just one, and the two remaining options are equally likely.

Source: Counterintuitive-Statistics

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6. More women filed for bankruptcy in 2003 than graduated from college. True or False.

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True. More women filed for bankruptcy in 2003 than graduated from college.

Source: Statistics on Women

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7. Are your lifetime odds of dying due to contact with hot tap water higher than dying due to a lightning strike?

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No, you are more likely to die of lightning strike. I had to through in something that I considered intuitive. Your lifetime odds of dying due to contact with hot tap water are 1 in 88,631. The odds for lightning are 1 in 79,399.

Sources: NSC

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8. OK, are you more likely to die due to contact with hot tap water or contact with a venomous snake?

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Hot tap water by a large amount. Lifetime odds of dying due to contact with venomous snake are 1 in 544,449. See above for hot water deaths.

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9. One more – more likely to die due to contact with hot tap water or from an earthquake?

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Hot tap water. Odds of dying in an earthquake is 1 in 100,004. Which do you worry about more?

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10. How many legs does an elephant have if you count his trunk as a leg?

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Come on now! Just because you mistakenly count a trunk as a leg, the trunk does not magically become a leg. Even if you count an elephant’s trunk as a leg, an elephant still only has 4 legs.

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