Probability
Wayne's Word Probability Page


Randomized Coin Flipper: 5050 Odds For Heads or Tails
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A Summary Of Elementary Probability
Five coins have a total of 32 permutations: H = Head & T = Tail


HHHTT

TTTHH





HHTTH

TTHHT





HHTHT

TTHTH





HTHTH

THTHT





HTHHT

THTTH




HHHHT

HTTHH

THHTT

TTTTH



HHHTH

TTHHH

HHTTT

TTTHT



HHTHH

THHHT

HTTTH

TTHTT



HTHHH

THTHH

HTHTT

THTTT


HHHHH

THHHH

THHTH

HTTHT

HTTTT

TTTTT






5 H's

4 H's 1 T

3 H's 2 T's

2 H's 3 T's

1 H 4 T's

5 T's

The above table of coin permutations is an example of Pascal's Triangle. It can be expressed algebraically by the following binomial expansion:
(H + T)^{5} =

H^{5}

+ 5 H^{4} T

+ 10 H^{3} T^{2}

+ 10 H^{2} T^{3}

+ 5 HT^{4}

+ T^{5}


5 H's

4 H's 1 T

3 H's 2 T's

2 H's 3 T's

1 H 4 T's

5 T's

The following questions refer to tosses involving all five coins:
1. What is the chance of getting all Heads (i.e. HHHHH)? A glance at the above Pascal's Triangle reveals only one permutation out of 32.
HHHHH = 1/2 X 1/2 X 1/2 X 1/2 X 1/2 = 1/32
2. What is the chance of getting 3 Heads and 2 Tails in that exact order (i.e. HHHTT)? As in the previous example there is only one permutation out of 32 (refer to the top permutation, 3rd column from left).
HHHTT = 1/2 X 1/2 X 1/2 X 1/2 X 1/2 = 1/32
3. What is the chance of getting 3 Heads and 2 Tails in any order? In this example you must consider all possible permutations with 3 Heads and 2 Tails. The 3rd column from left in the above Pascal's Triangle shows 10 permutations out of 32 with 3 Heads and 2 Tails. This is also the probability of having 3 girls and 2 boys when all possible orders are considered. Another way to solve this problem is to multiply 1/32 by the number of permutations: 1/32 X 10 = 10/32 = 5/16.
# of permutations: 5! / 3! 2! = 5 X 4 X 3 X 2 X 1 / 3 X 2 X 1 X 2 X 1 = 120 / 12 = 10
4. What is the chance of getting any permutation except 5 Heads?
5. What is the chance of getting at least one Head?
6. What is the chance of getting one Head?
7. What is the chance of getting no Heads?
8. What is the chance of getting at least 3 Heads?

Probability In Rolling A Pair Of Dice
Miniature dice on a U.S. penny. Each die is 4 mm on a side.
The probability for two 5's is 1/36 or 3/36 for total of 10 dots.
A total of 36 different combinations: 6 X 6 = 36
Die

R e d

G r e e n

Number of Dots








2

3

4

5

6

7


3

4

5

6

7

8


4

5

6

7

8

9


5

6

7

8

9

10


6

7

8

9

10

11


7

8

9

10

11

12

There are 36 possible combinations when rolling a pair of dice. Of all the combinations of dots, seven is the most likely number. The probability of coming up with a seven is 1/6 (one out of six rolls) because six combinations of red and green dice add up to seven (6/36 = 1/6). The chance of getting a two (snake eyes) is only 1/36.

Probability In Drawing A Royal Flush
Drawing Cards In Order From A Randomly Shuffled Deck
A royal flush is an ace, king, queen, jack and ten in the same suit. If the cards are drawn in that order from a deck of 52 cards, then the chance of the first card being an ace is 4/52 since there are four aces in the deck and no suit has been specified. The chance of the second card being a king of the same suit is 1/51 because there is only one king of that suit and there are 51 cards left in the deck. The following mathematical calculation shows the unlikely probability of drawing a royal flush in order from a deck of 52 cards:
4/52 X 1/51 X 1/50 X 1/49 X 1/48 = 4 / 311,875,200
[This is roughly one chance out of 78 million.]
It is more likely to draw a royal flush from a deck of 52 cards if the cards can be drawn in any order. In this case the first card is 20/52 because there are four suits (club, spade, heart & diamond) and five different cards (ace, king, queen, jack & ten) to choose from. Then you must consider all the possible orders (permutations) that the cards can be drawn, such as acekingqueenjackten, tenjackqueenkingace, acejacktenkingqueen, etc. The number of permutations can be calculated from five factorial or 5! = 5 X 4 X 3 X 2 X 1 = 120:
20/52 X 1/51 X 1/50 X 1/49 X 1/48 X 120 = 2400 / 311,875,200
[This is roughly one chance out of 130 thousand.]

Probability Of Winning The California Lottery
Picking Six Out Of 53 Numbers With A Single Ticket
The probability of one winner picking all six correct numbers out of 53 numbers is similar to the above question about drawing a royal flush from a deck of cards. The first number is 1/53 because it is one number out of 53 possibilities. The second number is 1/52 because the total number is now reduced by one to 52. Since there are different permutations for the six numbers, you must multiply the total probability by six factorial or 6! = 6 X 5 x 4 X 3 X 2 X 1 = 720:
1/53 X 1/52 X 1/51 X 1/50 X 1/49 X 1/48 X 720 = 720 / 16,529,385,000
[This is roughly one chance out of 23 million.]
According to B. Siskin and J. Staller (What are the Chances?, Crown Publishers, Inc., New York, 1989), the chance of being struck by lightning in your lifetime is one in 600,000. Compare this with the chance of winning the state lottery (one in 23 million)! Of course, the chance of being struck by lightning is increased significantly if you stand on a barren mountain summit (above timberline) holding a metal rod during a thunder storm.

Probability Of Sex Determination In Humans
Sex Determination Is More Complicated Than Tossing A Coin
Fertilization (syngamy) is the fusion of two haploid gametes (the sperm and the egg) to form a diploid (2n) zygote. This is how the chromosome number in a life cycle changes from haploid (n) to diploid (2n). Since human males produce Xbearing and Ybearing sperm, and human females produce only Xbearing eggs, the gametes combine randomly according to the following table:
Gametes

Xbearing sperm

Ybearing sperm

Xbearing egg

XX

XY

The male (XY) and female (XX) offspring in the above table are in a 5050 ratio with an equal number of boys and girls. Therefore, the chance of having a boy is 1/2 or 50% and the chance of a girl is also 1/2 or 50%. This ratio can be demonstrated by tossing a coin many times and keeping track of the number of heads and tails. If enough tosses are made, the number of heads and tails should be very close to 5050.
Unfortunately in biology, sex ratios in humans are not that easily explained. In the United States, there is a slightly better chance of having a boy, about 105 males to 100 females. There are a number of hypotheses (tentative explanations) for this unequal birth ratio, most of which are probably not accurate. If this unequal birth ratio is the result of a greater number of male conceptions, then perhaps the Ysperm has a slightly better advantage in reaching the egg or penetrating the barrier of follicle cells around the egg, possibly by the smaller size of its head and faster speed. According to some references, muscular contractions and ciliary currents within the female reproductive tract are primarily responsible for transporting the sperm to the egg. In vitro, sperm can swim about 3 mm per minute, but within the vagina and oviduct (in vivo) they can travel about 5 mm per minute. Since the speed is additive, a faster Ysperm could potentially still win the race to the egg. It has been demonstrated that Ybearing sperm do not live as long as Xbearing sperm, so the time of ovulation and sexual intercourse could be a factor in determining the sex of a child. By the time the sperm reach the upper part of the fallopian tube where fertilization occurs, most of the Ysperm may have already died. In this case when the egg is released from the ovary, the odds would be favor fertilization by an Xbearing sperm resulting in a girl. Another hypothesis for unequal sex ratios is an unequal number of Xbearing and Ybearing sperm in the man's semen. Some references state that this condition might be hereditary, but it is not clearly explained in more scholarly texts.
A more plausible explanation for unequal birth ratios may involve the rejection of an embryo by the mother's antibodies. Conceptions may be approximately equal, but a greater loss of female embryos early in the gestation period may account for slightly more male births. The mother has two Xchromosomes, one from her mother and one from her father. Perhaps the Xchromosome from her father carries a gene that sensitizes her against the female fetus. In other words she develops antibodies against proteins of the female fetus but not the male fetus. Remember that these are only hypotheses at this time and should not be accepted as the final answer. When more testing is done, perhaps one or more of these hypotheses will become a widelyaccepted theory.
Front Right: Rubik's Cube: Each of the 6 faces (sides) of cube contains 9 stickers, each one of 6 colors (red, green, blue, white, orange & yellow). A pivot mechanism enables each vertical or horizontal row of 3 stickers to turn independently, thus mixing up the colors. The object of puzzle is to return each side to a single color of 9 stickers. There are 8! (40,320) ways to arrange the corner cubes. 7 corner cubes can be oriented independently, and the orientation of the 8th depends on the preceeding 7, giving 3^{7} (2,187) possibilities. There are 12! /2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. 11 edges can be flipped independently, with the flip of the 12th depending on the preceeding ones, giving 2^{11} (2,048) possibilities.
Number of arrangements: 8! x 3^{7} x 12! /2 x 2^{11} = 43,252,003,274,489,856,000 or approximately 43 quintillion. According to Wikipedia, this number of Rubik's cubes could cover the earth's surface more than 275 times!

Puzzle Games Based On Arrangements (Permutations) Of The Sides Of Cubes
Instant Insanity: 4 cubes whose faces (sides) have 4 colors (red, green, blue & white). The object is to stack cubes so that all 4 colors show on each side of stack. Each cube can be arranged in 24 different positions (6 sides x 4 colors = 24). The number of possible arrangements is 24^{4} = 331, 776; however, the solved puzzle has multiple valid orientations so you must divide 331,776 by the number of valid orientations. See folowing:

According to Wikipedia, the solution is symmetrical 8 ways: If you have a solution, and you flip all of the cubes forward, you have another valid solution. You can do that move 4 times. You can also rotate each cube 180 degrees around its vertical axis, which gives 2 more possiblities, a total of 4 x 2 = 8. Therefore, the total number of arrangements (24^{4}) must be divided by 8. NOTE: As of 22 March 2023, I question previous calculation. See next paragraph with blue background:

After carefully examining my puzzle and the rules for puzzle solution, I am convinced there are 24 configurations for a vertical stack of 4 cubes, each side with 4 colors as specified in the rules. Each configuration can be rotated 4 times to verify the colors. If you randomly reshuffle the 4 cubes, you may get another stack with a different order of colors that counts as another solution, as long as all 4 sides have all 4 colors. Therefore, if you divide the total number of arrangements (24^{4}) by 24, the chance of getting a correct solution is 1/13,824. This is a little more like likely than 1/331,776!


24 Solution Configurations Instead Of 8 Described On Internet?
Not All Puzzles Are Identical!
To make the Instant Insanity explanation even more complicated, I discovered that I have at least 2 Instant Insanity puzzles with different color patterns on the cubes. For example, In the following image I have compared the rotated stacks of 2 different solved puzzles. The upper puzzle has a cube with 3 red sides. The lower puzzle has a cube with 3 green sides. The 2 puzzles have valid solutions with different color sequences.


