Sunday, February 22, 2009

6/49 Lotto: Probabilities, Odds, and Mathematical Expectation

Here are some hard facts about the 6/49 Lotto in terms of probabilities, odds, and mathematical expectation.

PRIZES AND PROBABILITIES

Jackpot (picking all the six winning numbers)

In choosing six out of the 49 numbers, there are C(49,6) ways. Using combination formula, this value is equal to 13,983,816. In picking all the six winning numbers, there is only one way, C(6,6). Just divide the two values obtained to determine its probability. That is, the probability of hitting the jackpot is 1/13,983,816 or 0.0000000715112. This is approximately the same as obtaining the same face 8 times in succession when tossing a fair die or getting 24 tails in succession when flipping a fair coin.
TRUE ODDS: 13,983,815 to 1

Second Prize (picking five winning numbers)

Since the five number chosen are winning numbers, the other number chosen must be from the 43 nonwinning numbers. This is equal to
C(6,5)xC(43,1) or 258. Hence, the probability of winning the second prize is 258/13,983,816 or 0.00001845. This is approximately the same as obtaining the same face 6 times in succession when tossing a fair die or getting 16 tails in succession when flipping a fair coin.
TRUE ODDS: 54,200.8 to 1

Third Prize (picking four winning numbers)

If the four numbers picked are winning numbers, the other two numbers must be from the 43 nonwinning numbers. This is equal to
C(6,4)xC(43,2) or 13, 545. The associated probability for this prize is 13,545/13,983,816 or 0.0009686, roughly the same as obtaining the same face 4 times in succession when tossing a die or 10 tails in succession when flipping a fair coin.
TRUE ODDS: 1,031.4 to 1

Fourth Prize (picking three winning numbers)

The number of ways in choosing six numbers three of which are winning numbers is equal to C(6,3)xC(43,3) or 246,820. The probability therefore of winning this prize is 246,820/13,983,816 or 0.0177, roughly the same as getting 6 tails in succession when flipping a fair coin.
TRUE ODDS: 55.7 to 1

MATHEMATICAL EXPECTATION

In computing the mathematical expectation of this game, the probability of NOT winning any of the prizes must be computed. It is obtained by subtacting the sum of the probabilities of winning at least one of the prizes from 1. Its probability is
13,723,192/13,983,816 or 0.981362455.

Aside from knowing the probability of not winning, it is likewise important to know the prizes. Assuming that the jackpot prize is Php 16,000,000, the second prize is Php 56, 000, the third prize is Php. 1,000, and the fourth prize is Php 100, then the mathematical expectation is equal to Php 16,000,000(1/13,983,816) + Php 56,000(258/13,983,816) + Php 1000(13,545/13,983,816) + Php 100(246,820/13,983,816) + 0(13,723,192/13,983,816) or Php 4.91. This means that for every 20-peso ticket, one should expect to win only Php 4.91. (From a pessimist's point of view, one should expect to lose Php 15.09)

JACKPOT PRIZE: MATHEMATICAL EXPECTATION=TICKET PRICE

The mathematical expectation computed previously assumes that the jackpot's Php 16,000,000. In lottery, however, when no one matches the winning numbers, a jackpot rolls over and is added to the next drawing. At what prize then is the mathematical expectation equal to the ticket's worth? For a ticket worth Php 20, the jackpot prize should be Php 227,001, 320. (It is assumed here that only one ticket is taken, only one wins the jackpot, and prizes are tax-free.)

FAVORABLE TIME TO BET

The favorable time to buy a ticket in 6/49 lotto, mathematically speaking, is when the jackpot prize exceeds Php 227M. This would simply mean that the mathematical expectation is higher than the ticket price.

Thursday, February 19, 2009

Chasing Tails

A big cat saw a little cat chasing its tail and asked, “Why are you chasing your tail?”

Said the kitten, “I have learned that the best thing for a cat is happiness and that happiness is in my tail. Therefore, I am chasing it: and when I catch it, I shall have happiness.”

Said the old cat, “My son, I, too, have paid attention to the problems of the universe. I, too, have judged that happiness is in my tail. But, I have noticed that whenever I chase it, it runs away from me and when I go about my business, it just seems to come after me wherever I go.”

"On Happiness", C. L. James

Saturday, February 14, 2009

Proving 3 = 2 and 4 = 3

One way of grabbing students' attention, according to John Keller, the proponent of the ARCS Model of Motivational Design, is to use incongruity and conflict. This can be done by presenting students with statements that go against their previous knowledge. Here are two fallacies that will surely catch students' attention.

(1) Prove that 3 = 2.

Proof:
Let a and b be equal non-zero quantities:
a = b

Multiply both sides by a2:
a3 = a2b

Subtract b3:
a3 - b3 = a2b - b3

Factor both sides:
(a - b)(a2 + ab + b2) = b(a2 - b2)
(a - b)(a2 + ab + b2) = b(a + b)(a - b)

Divide both sides by (a - b):
a2 + ab + b2 = b(a + b)

Multiply terms on the right side:
a2 + ab + b2 = ab + b2

Since a = b, both sides may be expressed in terms of b:
b2 + b2 + b2 = b2 + b2

Combine like terms on both sides:
3b2 = 2b2

Divide by the non-zero b2:
3 = 2

Q.E.D.

(2) Prove that 4 = 3.

Proof:
Let a and b be equal non-zero quantities:
a = b

Multiply both sides by a3:
a4 = a3b

Subtract b4:
a4 - b4 = a3b - b4

Factor both sides:
(a2 + b2)(a2 - b2) = b(a3 - b3)
(a2 + b2)(a + b)(a - b) = b(a - b)(a2 + ab + b2)

Divide both sides by (a - b):
(a2 + b2)(a + b)= b(a2 + ab + b2)

Multiply terms on both sides:
a3 + ab2 + a2b + b3 = a2b + ab2 + b3

Since a = b, both sides may be expressed in terms of b:
b3 + b3 + b3 + b3 = b3 + b3 + b3

Combine like terms on both sides:
4b3 = 3b3

Divide by the non-zero b3:
4 = 3

Q.E.D.

Wednesday, February 11, 2009

Grammy Song of the Year

American Boy by Estelle and Kanye West, Chasing Pavements by Adele, I'm Yours by Jason Mraz, Love Song by Sarah Bareilles, and Viva la Vida by Coldplay were nominated for this year's Grammy Song of the Year. The last one won the award.

VIVA LA VIDA by Coldplay

I used to rule the world
Seas would rise when I gave the word
Now in the morning I sleep alone
Sweep the streets I used to own

I used to roll the dice
Feel the fear in my enemy's eyes
Listen as the crowd would sing
"Now the old king is dead! Long live the king!"

One minute I held the key
Next the walls were closed on me
And I discovered that my castles stand
Upon pillars of salt and pillars of sand

I hear Jerusalem bells a ringing
Roman Cavalry choirs are singing
Be my mirror, my sword and shield
My missionaries in a foreign field

For some reason I can't explain
Once you go there was never
Never an honest word
And that was when I ruled the world

It was the wicked and wild wind
Blew down the doors to let me in
Shattered windows and the sound of drums
People couldn't believe what I'd become

Revolutionaries wait
For my head on a silver plate
Just a puppet on a lonely string
Oh who would ever want to be king?

I hear Jerusalem bells a ringing
Roman Cavalry choirs are singing
Be my mirror, my sword and shield
My missionaries in a foreign field

For some reason I can't explain
I know Saint Peter won't call my name
Never an honest word
But that was when I ruled the world

I hear Jerusalem bells a ringing
Roman Cavalry choirs are singing
Be my mirror, my sword and shield
My missionaries in a foreign field

For some reason I can't explain
I know Saint Peter won't call my name
Never an honest word
But that was when I ruled the world

Saturday, February 7, 2009

Commonly Mispronounced Math Words

Here is a list of words commonly used in mathematics which are oftentimes mispronounced by students and even by teachers. Included in the list are the subjects where the words are usually encountered. Just click the word to know its meaning and pronunciation.

1. annulus (Geometry)
2. aliquot (Arithmetic)
3. Cartesian (Algebra, Geometry)
4. chi (Inferential Statistics)
5. coplanar (Geometry)
6. cosecant (Trigonometry)
7. echelon (Matrix Algebra)
8. Euclidean (Geometry)
9. Euler (Geometry, Number Theory)
10. Fibonacci (Number Theory)
11. foci (Analytic Geometry)
12. Lie Algebra
13. loci (Analytic Geometry)
14. parallelepiped (Solid Geometry)
15. pint (Measurement)
16. phi (Number Theory)
17. polyhedron (Solid Geometry)
18. Pythagorean (Plane Geometry)
19. quotient (Arithmetic)
20. Riemannian (Modern Geometry)
21. secant (Trigonometry)
22. scalar (Linear Algebra)
23. tableau (Matrix Algebra)
24. theta (Trigonometry)
25. trichotomy (Algebra)
26. Trigonometry
27. variability (Descriptive Statistics)
28. variable (Algebra)