Probability

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Probability

TAKING A CHANCE...

PROBABILITY
Now that you know all about how to count, you can compute probabilities.
A probability is a measure of chance.
A probability is a number between 0 and 1 inclusive.
The bigger the number, the better the chance.
The smaller the number, the less of a chance.
A probability is the quotient of two counts:
The first count is the number of ways something can happen that you want
divided by the total number of ways something can happen whether you like it or not.
We can concoct a probability formula like this;

P(x) = WAYS SPECIFIC EVENTS CAN HAPPEN / WAYS ALL EVENTS CAN HAPPEN
P(x) = WAYS WANTED / WAYS TOTAL
P(x) = YOUR COUNT / TOTAL COUNT
P(x) = SOME WAYS / ALL WAYS,

where P(x) means the probability of an event x.
For example: what is the probability of drawing an Ace from a deck of cards.
There are 4 ways to draw an Ace from a deck of cards with 52 cards
4 ways you want and 52 ways total.
Therefore, P(Ace) = 4/52 = .077 = 77%

Note that a probability may be written as a fraction, decimal, or percentage.

There are thus 48 ways not to draw an Ace against 4 ways to draw one.
We call these the Odds of drawing an Ace.
We write Odds like this 48 to 4 or 48:4.

Since the number of some is always less than or equal to the number of all,
probability is always less than or equal to 1,
but probability cannot be negative either.

For more about probability,
read the Wikipedia on Probability.

QUESTIONS — PART 1

During a given year, about 20 out of every 100 firefighters are injured at work. The probability of a firefighter being injured during a given year is about or 20/100 or 1/5. As a percentage, this probability is 1/5 x 100 = 20%. What is the probability that a fireman will NOT be injured at work?
If a frog catches a fly, the probability that the fly was alive when the frog ate it is 100%.
1. What does that mean?
2. If you touch a toad, the probability that you will get warts as a result is 0%. What does that mean?
Probabilities about smoking are listed below:
- A smoker wants to quit 66%
- A smoker has tried to quit 84%
- A smoker will succeed in quitting 21%
1. Do most people who smoke want to quit?
2. What is the probability that someone who smokes does not want to quit?
3. What is the probability that someone who smokes has never tried to quit?
4. Do most people who try to quit smoking succeed?
5. What is the probability that someone who tries to quit smoking will not succeed?
When an ordinary die is thrown, it is equally likely that any one of its six faces will turn up. If it is thrown once, what is the probability
1. of getting a 3?
2. of getting a 4?
3. of an odd number turning up?
A die is misspotted with two faces marked with 1 spot, two faces marked with 4 spots, and two faces marked with 5 spots. If it is thrown once, what is the probability
1. of getting a 3?
2. of getting a 4?
3. of an odd number turning up?
A multiple-choice test has five choices for each answer. If you have no idea of which answer to a question is correct, what is the probability
1. of guessing the correct answer?
2. of guessing a wrong answer?
If you realize that two of the five answers to a question are not correct, what is the probability
1. of guessing the correct answer for that question?
2. of guessing a wrong answer?
Some holidays always occur on the same day of the week and other holidays do not. If a year is chosen at random, what is the probability that
1. the Fourth of July is on a Saturday?
2. Thanksgiving is on a Thursday?
3. Easter is not on a Sunday?
4. Halloween is not on a Friday?
Suppose that 52 cards are shuffled and turned over. If one card is then chosen at random, what is the probability that the card is
1. a jack?
2. a heart?
3. a face card?
4. the queen of diamonds?
5. not the queen of diamonds?
Out of every 80 telephone calls attempted, 8 result in busy signals and 10 result in no answer.
1. Which is more probable: getting a busy signal or getting no answer?
2. What is the probability of getting a busy signal as a fraction?
3. What is the probability of getting a busy signal as a percentage?
4. What is the probability of getting no answer as a fraction?
5. What is the probability of getting no answer as a percentage?
Probabilities for various events occurring in major league baseball games are listed below.
- The losing team doesn�t score 0.15
- Both games of a double-header are won by the same team 0.47
- The winning run is scored in the last inning 0.19
- No bases are stolen during the game 0.36
- A game goes into extra innings 0.09
- The home team wins the game 0.53
- A game has at least one home run 0.64
1. Which one of these events is the most likely to occur?
2. Which one is the least likely to occur?
3. Which two are the most unpredictable?
In a shell game, the operator covers a pea with one of three shells and then moves the shells about. The player tries to keep track of the pea and then bets the operator that he knows where it is. Suppose that the operator moves the shells so rapidly that the player cannot follow them and must guess where the pea is.
1. If the game is an honest one, what is the probability that the player will guess the correct shell?
2. If the game is honest and you played it six times, how many times would you expect to win?
3. Shell games are usually not operated honestly. What do you suppose is the probability that the player will win in a dishonest game?
Suppose that an ordinary coin is tossed. If it is assumed that the outcomes of heads and tails are equally likely, then the probability of it turning up heads is 1/2, or 0.5.
1. By this reasoning, what is the probability of a coin turning up tails?
2. An English mathematician being held prisoner during World War II tossed a coin 10,000 times. The coin turned up heads 5,067 times. Use these numbers to calculate the probability of the coin turning up heads.
3. By the same method, what is the probability of the coin turning up tails?
4. Suppose that the probability of a certain coin turning up heads is 1. What would you conclude about the coin?
If someone in New York City is treated in an emergency room for a bite,
1. The probability that the person was bitten by a dog is 9/10,
2. The probability that the person was bitten by a cat is 1/20,
3. The probability that the person was bitten by another person is 1/25.
4. Express these probabilities as percentages.
5. Do the three percentages that you have just determined add up to 100%?
6. Do you think they should?
7. Explain why?
An American roulette wheel has 38 compartments around its rim. Two these are numbered 0 and 00 and are green; the others are numbered from 1 to 36, of which half are red and half are black. When the wheel is spun in one direction, a small ivory ball is rolled in the opposite direction along its rim. If the wheel is a fair one, the chances of the ball falling into any one of the 38 compartments as it slows do are equally likely. Some typical bets in roulette are listed below. Express the probability of winning each bet both as a fraction and as a percentage.
1. The number 7.
2. A black number.
3. An odd number.
4. The numbers 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, and 36 (called a �column� bet).
5. Two adjoining numbers (called a "split" bet).
6. A red number, if all 26 numbers that had come up previously were black.
7. Which one of these bets do you think has the biggest payoff?
What is the probability of winning in the Lucky 4 game on the combinations page?

TEXAS HOLD'EM — PART 2

In Texas Hold'em, you and the other players are each dealt two hole cards that no one else sees. Five cards are then dealt face down in the middle of the table for everyone to use to make standard poker hands. These are called the community cards. The first three community cards are called the flop. The fourth community card is called the turn. The fifth community card is called the river.

If you have exactly one Ace in the hole, what is the chance that the community cards will contain NO Aces? Ask these questions when counting the ways one of the Aces may be contained in the community cards:
- At the start of each hand, how many cards are unknown to you?
- Of these unknown cards, how many subsets of five community cards are possible? Notice that this question asks for subsets, not arrangements of the community cards.
- How many Aces remain unseen by you?
- How many ways may the first community card NOT be an Ace?
- How many ways may the second community card NOT be an Ace?
- How many ways may the third community card NOT be an Ace?
- How many ways may the fourth community card NOT be an Ace?
- How many ways may the fifth community card NOT be an Ace?
- If you are counting subsets, does the order matter?
- After you calculate the ways an Ace may NOT be contained among the community cards, divide the count by the total possible subsets of community cards.
If you have exactly one Ace in the hole, what is the chance that the community cards will contain at least one Ace also? Ask these questions when counting the ways an Ace may be contained in the community cards:
- If you know the number of ways an Ace would not be included in the community cards, how would you find the number of ways an Ace would be included in the community cards?
- Divide this number by the total possible subsets of community cards.
- A quicker way would be to subtract the probability calculated in the previous question from 1. Why? Because these are complementary probabilities. That is, it either happens one way or the other. Therefore, the sum of their probabilities must equal 1.
If you have two Aces in the hole, what is the chance that the community cards will contain NO Aces?
- Once again ask how many Aces remain unseen by you?
- Follow the procedure in question 1.
- Notice that the numbers you use change slightly.
If you have two Aces in the hole, what is the chance that the community cards will contain at least one Ace also?
- Follow the procedure in question 2, OR...
- Find the complementary probability calculated in the previous question.
The flop is revealed and it contains no aces. What is the probability now that one of the two remaining community cards contains an Ace to match your Ace in the hole?
- At this point you know where 5 cards are. How many do you not know?
- Of these cards you do not know, how many total ways can two of them make subsets of size 2?
- Now count the ways those two cards cannot contain an Ace.
- Subtract these ways from the total ways. These are the ways you want.
- Divide the ways you want by the total ways.
Suppose four of the community cards are revealed and they contain no Aces. If you have exactly one Ace in the hole, what is the chance that the last remaining community card is an Ace?
- How many Aces are left? You want the last card to be one of them.
- How many total cards do you not know?
- Divide the number of ways you can get what you want by this total.
If your hole cards are the King and Queen of Hearts, and the flop contains the Four of Clubs and the Two and Three of Hearts, what is the chance that at least one of the two remaining community cards will be a Heart?
- At this point you know where 5 cards are. How many do you not know?
- Of these cards you do not know, how many total ways can two of them make subsets of size 2?
- Now count the ways those two cards cannot contain a Heart.
- Subtract these ways from the total ways. These are the ways you want.
- Divide the ways you want by the total ways.
Suppose four of the community cards are revealed and they contain exactly two Hearts. If your hole cards are the King and Queen of Hearts, what is the chance that the last remaining community card will be a Heart?
- How many Hearts are left? You want the last card to be one of them.
- How many total cards do you not know?
- Divide the number of ways you can get what you want by this total.
If your hole cards are the King and Queen of Hearts, and the flop contains the 2 of Spades, Jack of Clubs, and 10 of Diamonds, what is the chance that at least one of the two remaining community cards will be a 9 or an Ace?
- At this point you know where 5 cards are. How many do you not know?
- Of these cards you do not know, how many total ways can two of them make subsets of size 2?
- Now count the ways those two cards cannot contain a 9 or an Ace.
- Subtract these ways from the total ways. These are the ways you want.
- Divide the ways you want by the total ways.
Suppose four of the community cards are revealed and they contain no 9's or Aces. If your hole cards are the King and Queen of Hearts, what is the chance the last remaining community card will be a 9 or an Ace?
- How many 9's and Aces are left? You want the last card to be one of them.
- How many total cards do you not know?
- Divide the number of ways you can get what you want by this total.

Complete the table below for questions 1 thru 10 above.
Probabilities must be precise to 3 significant digits.

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Last Modified 10/23/08 8:34 AM