Foundations ofMathematics

Week 7: Probability

Subjective, empirical and theoreticalprobability

David RadcliffeOctober 28, 2005

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What is probability?

• Probability is a mathematical tool for making decisions in an uncertainworld.

• The probability of an event is a number between 0 and 1 that expresseshow likely it is to occur.

• Probability 0 means that the event is impossible.

• Probability 1 (or 100%) means that the event is certain.

• The greater the probability, the more likely it is to occur.

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Three approaches to probability

• Subjective probability is based on a person’s belief that an event willoccur.

• Empirical probability (or experimental probability) is based on howoften an event has occurred in the past.

• Theoretical probability is based on a mathematical model ofprobability.

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Subjective probability

• Subjective probability measures a person’s belief that an event will occur.

• These probabilities vary from person to person.

• They also change as one learns new information.

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Example of subjective probability

Two weeks ago, Sen. Richard Lugar (R., Ind.), chairman of theSenate Foreign Relations Committee, released the results of anambitious survey of arms experts. The study was conducted in late2004 and early 2005. On average, the 85 respondents predicted a29.2% chance of a nuclear attack in the next decade, with 79% sayingthat such an attack was more likely to be carried out by terroriststhan by a government. Sen. Lugar said in the report that “theestimated combined risk of a WMD attack over five years is as highas 50%. Over 10 years this risk expands to as much as 70%.”

(Source: WSJ, July 7, 2005)

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Survey question

Average retail gas prices in the United States rose above $3 per gallon in theaftermath of Hurricane Katrina. What is your estimate of the probabilitythat gas prices will again reach $3 per gallon in the next six months?

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Probability and betting• Subjective probability reflects how much one would be willing to bet on

an outcome.

• Example: A person should be willing to pay up to $30 for the chance towin $100, if the probability of winning is 30%.

• A project should be accepted if its cost is less than the benefit times theprobability of success.

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Exercise

You are the CEO of a corporation. A project has been proposed to develop anew product. If successful, the project will generate $10 million in revenue.The cost of the project is $3 million. However, the probability of success isestimated at only 40%. Should you approve the project?

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Exercise

You are the CEO of a corporation. A project has been proposed to develop anew product. If successful, the project will generate $10 million in revenue.The cost of the project is $3 million. However, the probability of success isestimated at only 40%. Should you approve the project?

Answer: Yes, because the cost is less than 40% of the anticipated revenue.

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Empirical probability

Empirical (or experimental) probability is determined by how often an eventhas been observed to occur.

Empirical probability =# of successes

# of trials

Empirical probability only makes sense if the event is repeatable.

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Exercise

In the 2004-05 season, Kobe Bryant made 573 field goals in 1324 attempts.What is the empirical probability that Kobe will sink a field goal?

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Exercise

In the 2004-05 season, Kobe Bryant made 573 field goals in 1324 attempts.What is the empirical probability that Kobe will sink a field goal?

Answer:

5731324

= 0.433, or 43.3%.

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Application: Benford’s Law

Benford’s law states that in lists of numbers from many real-world sources ofdata, the leading digit 1 occurs more frequently than the other digits – about30% of the time.

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Theoretical probability• A random experiment is an act whose outcome is uncertain.

• The sample space S is the set of all possible outcomes of the randomexperiment.

• Each outcome is assigned a probability between 0 and 1. Theprobabilities of all the outcomes add up to 1.

• An event is a subset of the sample space (i.e. it is a set of outcomes).

• The probability of an event is the sum of the events of the outcomes thatcomprise the event.

• We write P (E) to denote the probability of the event E.

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Example• Consider the random experiment of tossing a fair coin twice.

• The sample space is S = {HH, HT, TH, TT}

• Since the coin is fair, P (HH) = P (TH) = P (TH) = P (TT ) =14

• The event of getting exactly one head is {HT, TH}.• The probability of that event is

P ({HT, TH}) = P (HT ) + P (TH) =14

+14

=12

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ExerciseAn urn contains a large number of ping-pong balls. Half of the balls arewhite, 30% of the balls are red, and the rest of the balls are blue. Thefollowing random experiment is conducted: a ball is removed from the urn,and its color is recorded.

1. What is the sample space of this experiment?

2. What is P (white)?

3. What is P ({red,white})?4. What is P (blue)?

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Equiprobable spaces• A sample space is said to be equiprobable if all outcomes have the same

probability.

• If the space is equiprobable, then each outcome has probability 1/n,where n is the size of the sample space, and the probability of an eventwith k outcomes is k/n.

Pr(E) =#E

#S

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Exercise

A poker deck has 52 cards, and 4 of these cards are aces. A card is drawn atrandom:

1. What is the probability that the card is an ace?

2. What is the probability of drawing the ace of spades?

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Exercise

A poker deck has 52 cards, and 4 of these cards are aces. A card is drawn atrandom.

1. What is the probability that the card is an ace? 4/52

2. What is the probability of drawing the ace of spades? 1/52

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Review Questions1. What is the probability that the sun will rise tomorrow? What is the

probability that the sun will rise in the West?

2. What is the difference between subjective and empirical probability?

3. Identify each of the following statements as subjective, empirical, ortheoretical probabilities.

(a) The probability of being dealt a full house in poker is 1 in 693.

(b) There is a 40% chance of rain tomorrow.

(c) Avalanche victims who are found within the first 15 minutes have a92 percent chance of survival.

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Review Questions II4. A man buys a ticket for a raffle with a $10,000 prize. The probability of

winning is 0.05%. How much (in theory) should he pay for the ticket?Why might he be willing to pay more than this?

5. An oil company plans to invest $5 million to explore a potential oil field.If successful, the project will generate $25,000,000 in revenue. Whatprobability of success is required to justify this project?

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Review Questions III6. A coin is tossed 100 times. It lands heads 61 times, and tails 39 times.

(a) What is the empirical probability that the next toss will land heads?

(b) What is the theoretical probability that the next toss will land heads,assuming that the coin is fair?

7. What is the probability that Christmas will fall on a weekend?

8. Rolling a fair die is an example of a random experiment. What is thesample space of this experiment?

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