  Assignment 78

 "God is like a skillful geometrician." - (Sir Thomas Brown, 1605-1682) Morris Kline (passed away in the 1990's): Professor Emeritus of Mathematics at New York University, Kline exerted considerable influence on modern mathematical thought. His famous book, Why Johnny Can't Add , severely criticized the "new math" that swept the U.S. in the 1960's. A highly respected mathematics historian, he also authored Mathematics in Western Culture, Mathematical Thought from Ancient to Modern Times, and Mathematics: The Loss of Certainty. All are well worth reading! Kline's books treat the fundamental changes that humans have been forced to make in attempts to understand the nature and role of mathematics. The books point out that today there is not one universally accept concept of mathematics, but rather many conflicting one. Yet the effectiveness of mathematics in describing and exploring physical and social phenomena continues to expand. What did Herkimer use when he was helping the government take a census of all monkeys in zoos? Answer: An ape recorder. Herky wants to know: If you need to pass a civil service exam to work for the government, why don't taxpayers have to take the test? If a salesman tells you that a purchase will "pay for itself in no time," is it OK to ask him to send it to you when it does. ASSIGNMENT #78 Reading: Section 12.4, pages 724-727. Written: Page 727-728/29-40, 43, 44, 46, 47. Mathematical numbers:DEFECTIVE NUMBER: An integer number is defective if the sum of its proper divisors (all divisors except the number itself) is less than the number. The proper divisors of 32 are 1,2,4,8,16. Since 1+2+4+8+16 = 31, which is less than 32, the number 32 is a defective number. All prime numbers are defective numbers.
The probability of compound events is important (and highly dependent on you understanding basic language.)

Here's a simple example involving compound events.

Suppose I roll a single die: The only possible outcomes are 1,2,3,4,5,6. all equally likely.

Let A be the event "I get an even number."
Let B be the event "I get a number greater than 4."

What is the probability you get an even number OR a total greater than 4? In other words, what is the Prob(A or B)?

OK, we could just look at the outcomes that make up the event (A or B). These would include 2, 4, 5, 6. We could correctly conclude that Prob(A or B) = 4/6 = 2/3.

Now, Prob(A) = 3/6 and Prob(B) = 2/6. Note that Prob(A) + Prob(B) = 5/6, which does not equal Prob(A or B) = 2/3. The problem is that the outcome "6" is included in both events A and B, and has hence been counted twice when we compute Prob(A) + Prob(B). Now, Prob(A and B) = 1/6. We can now illustrate the formula

Prob(A or B) = Prob(A) + Prob(B) - Prob(A and B).

In this example, we have Prob(A or B) = 3/6 + 2/6 - 1/6 = 4/6 = 2/3, which agrees with our original computation.

The events A and B, as described, are not mutually exclusive. That is, there is something "common" to both events. In this case, the common outcome is 6, which is both an even number and a number greater than 4.

Here's another example: Suppose I randomly pick a card from a standard deck of 52 cards. What is the probability that I pick an ace or a heart?

OK, using the compound probability formula,

prob(ace or heart) = prob(ace) + prob(heart) - prob(ace and heart)

= 4/52 + 13/52 - 1/52 = 16/52, or about 30.77%.

Let's use plain old common sense. There are 13 hearts and four aces. Now, the ace of hearts is both a heart and an ace. Hence, the 13 hearts and the ace of spades , the ace of clubs, and the ace of diamonds represent the only 16 cards that make up the event "ace or heart." The desired probability is thus (13+3)/52 = 16/52, as determined by the formula.

 Problem: A card is randomly chosen from a standard deck of 52 cards. What is the probability that the card is a spade or a face card (a jack, queen, or king)? Solution (with communication): Probability (spade) = 13/52, probability (face card) = 12/52, andprobability (spade and face card) = 3/52. Hence probability(spade or face card) = 13/52 + 12/52 - 3/52 = 22/52 = 0.4231, or 42.31%. Problem: If I roll a single die, what is the probability that I get (a) a 5 or a 6? (b) a 5 or an odd number. Solution (with communication): (a) Since obtaining a 5 and obtaining a 6 are mutually exclusive events, the requested probability is 1/6 + 1/6 = 1/3. (b) The events getting a 5 and obtaining an odd number are not mutually exclusive. Probability(5) = 1/6, probability(odd number) = 3/6, and probability (5 and odd number) = 1/6. The desired probability is therefore 1/6 + 3/6 - 1/6 = 3/6, or 50%. (We might note that obtaining an odd number includes obtaining a 5. If you roll an odd number, you have the event described by obtaining a 5 or obtaining an odd number.)