Statistics Introduction

The study of probability is often deceptive: on the surface, it seems close to everyday experience and intuition seems enough to find answers to problems.

Terms such as "randomness," "chance," and so forth are used by laypeople such as the media, to justify one action over another.

The mathematical notion of probability is a different case. Terms are well-defined, rules formulated and proven, reason preferred over intuition.

Question One "A truth serum given to a suspect is known to be 90% reliable when the person is guilty. If you are guilty and you are given the serum, what is the probability that you will go free?”

From common sense, you would know the answer is 10%.

Question Two "A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?”

This needs some careful thinking and consideration but can be solved. "A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?”

This version cannot be solved! A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. If the suspect was selected from a group of suspects of which only a few of which have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?

There is a contradiction in the question A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime. Furthermore, 1% of the guilty are judged innocent by the serum. The serum indicates that he is guilty, what is the probability that he is innocent?

One part is irrelevant to answering the question A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. 25% of the suspects are scared to die. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?

Subtle irrelevancy A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. 25% of the suspects are scared to die. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime and only 1% have ever been found guilty, and the serum indicates that he is guilty, what is the probability that he is innocent?

Reword (in your head) to get the correct information "A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. This means that 90% of the guilty ones are judged guilty by the serum, and 10% of the guilty ones are judged innocent. Also, this means that 99% of the innocent ones are judged innocent, and 1% of the innocent ones are judged guilty. The suspect was selected from a group of suspects of which only 5% have ever committed a crime--this means that 5% are guilty. If the serum indicates that the suspect is guilty, what is the probability that he is innocent?"

Question Three Based on a conversation with somebody you meet for the first time you discover that this person has at least one son. You subsequently discover that this person has two children. What is the probability that the other child is a boy?

Answer First child Second child B G

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25%

Question Four The confusion results from two sources If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% The confusion results from two sources 1) There are percentage signs in the answers 2) The fact that 50% of the answers are “25%” and 25% of the answers are “50%”

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% The question does not say that the answer must among the listed options. It only asks what is the probability that you will be correct if  you answer at random.

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% The question instructs to only choose one single answer out of four. And assume a uniform distribution, since that is most likely intended, then each answer has a chance of 25% to become chosen.

Question Four So the correct answer should be: 25%. If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% So the correct answer should be: 25%.

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% This computes to answer A being correct, as well as answer D. Could that be?

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% Yes, it can. The question does not reveal how many of the four given answers are correct, but since there is one to be picked, assume that at least one of the four answers is correct.

Question Four Let's call answer A + answer D the correct answer pair. If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% Let's call answer A + answer D the correct answer pair.

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% Now, there are two possible choices (A or D) that result in 50% of the correct answer (A and D). Secondly, there is 50% chance of picking one (A or D) of two (A and D) out of four (A to D).

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% So whether answer A or answer D is chosen, in either case the probability of being correct (50% × 50%) is 25%, which evaluates true.

Question Four Thus, yes, the question has 2 correct answers. If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% Thus, yes, the question has 2 correct answers.

Question Four Do you agree with this logic? If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% Do you agree with this logic?

Question Four The question starts: If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% The question starts: "If you choose an answer to this question at random,

Question Four However it does not then continue: If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% However it does not then continue: "what is the probability that the answer chosen will be the probability of choosing that answer?”

Question Four It instead says: If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% It instead says: "what is the probability that you will be correct?" And then doesn't define correct.

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% However, lets presume that one of the following answers can be chosen: a, b, c or d The probability of choosing each answer:

Question Four a - 25% b - 25% c - 25% d - 25% 25% - 50% 50% - 25% If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% a - 25% b - 25% c - 25% d - 25% 25% - 50% 50% - 25% 60% - 25%

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% Being correct for "what is the probability that you will be correct?" if there is one correct answer (although as covered above the question doesn't define correct or specify how many answers are correct). This produces the same answer as the question "is the answer you choose the probability of choosing that answer?”

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% a – yes b - no c - no d - yes 25% - yes 50% - no 60% - no

Question Four If you choose an answer to this question at random, what is the chance you will be correct? A. 25% B 50% C 60% D 25% Could this be the real question in the question. "what is the probability that the answer chosen will be the probability of choosing that answer?": In which case, ‘b’ is correct!

Does this help? If you choose an answer to this question at random, what is the chance you will be correct? A) 250 B) 500 C) 600 D) 250

What if we start with this? If you choose an answer to this question at random, what is the chance you will be correct? A B C D

What if we start with this? If you choose an answer to this question at random, what is the chance you will be correct? A B C D Provided that one of the four option is correct (assumption), the probability will be 25% that you are correct.

What if we start with this? If you choose an answer to this question at random, what is the chance you will be correct? A B C D Since you are picking up a random answer (don't bother about logic at all), the correct answer to the question "what is the probability that you will be correct" is 25%. Do not bother about the options, which is misleading.

What if we start with this? If you choose an answer to this question at random, what is the chance you will be correct? A B C D But if there are two choices of 25% which is technically wrong because every choice must be different from the another.

What if we start with this? If you choose an answer to this question at random, what is the chance you will be correct? A B C D Chances of each selection at random are 1/4 or 25%. That means 25% would select choice A at random, 25% choice B at random and so on. Since we know choice A and D are the correct answer (they are repeated which is wrong), that leads us to 50% correct answer if people make random choices.

What if we start with this? If you choose an answer to this question at random, what is the chance you will be correct? A B C D So what is the probability that you will be correct is 50%

Each of the following statements is either true or false Each of the following statements is either true or false. Which of them are true and which are false? All of these sentences are false. Exactly 1 of these sentences is true. Exactly 2 of these sentences are true. Exactly 3 of these sentences are true. Exactly 4 of these sentences are true.

Sample question Imagine you have been tested in a large-scale screening programme for a disease known to affect one person in a hundred. The test is 90% accurate, and you test positive. What is the probability that you have the disease?

True or False

The squares marked A and B are the same shade of gray, yet they appear different

True!

Question Two http://www.agenarisk.com/resources/probability_puzzles/Making_sense_of_probability.html

http://web.mit.edu/persci/gaz/gaz-teaching/index.html

Monty Hall Problem http://www.grand-illusions.com/simulator/montysim.htm

http://www.math.ucsd.edu/~crypto/Monty/monty.html

Birthday Problem http://www-stat.stanford.edu/~susan/surprise/Birthday.html

Suppose a crime has been committed and that the criminal has left some physical evidence, such as some of their blood at the scene. Suppose the blood type is such that only 1 in every 1000 people has the matching type. http://www.agenarisk.com/resources/probability_puzzles/prosecutor.shtml

A suspect, let's call him Fred, who matches the blood type is put on trial. The prosecutor claims that the probability that an innocent person has the matching blood type is 1 in a 1000 (that's a probability of 0.001). Fred has the matching blood type and therefore the probability that Fred is innocent is just 1 in a 1000.

But the prosecutor’s assertion, which sounds convincing and could easily sway a jury, is wrong.

A fair dice with 6 sides is rolled a certain number of times and the number 1-6 is recorded each time it is rolled. Which of the following sequences (exact order) has the greatest probability of occurring? A) 12345 B) 654321 C) 2222 D) 241523

What were your assumptions? A die is rolled, find the probability that an even number is obtained. What were your assumptions?

What were your assumptions? Two coins are tossed, find the probability that two heads are obtained. What were your assumptions? Did you make more than one assumption?

Answer the question completely! Two dice are rolled, find the probability that the sum is a) equal to 1 b) equal to 4 c) less than 13

And again! A die is rolled and a coin is tossed, find the probability that the die shows an odd number and the coin shows a head.