Section 5.1 Day 2

1st Hour: 237 heads

3rd Hour: 232 heads

4th Hour: 208 heads

5th Hour: 200 heads

4. Suppose you spin a penny three times and record whether it lands heads up or tails up. (a) How many possible outcomes are there?

4. 8 HHH, HHT, HTH, HTT, THH, THT, TTH, TTT Suppose you spin a penny three times and record whether it lands heads up or tails up. (a) How many possible outcomes are there? 8 HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

4. (a) How many possible outcomes are there? 8 HHH, HHT, HTH, HTT, THH, THT, TTH, TTT (b) Are these outcomes equally likely? If not, which is most likely? Least likely?

4. (a) How many possible outcomes are there? 8 HHH, HHT, HTH, HTT, THH, THT, TTH, TTT (b) Are these outcomes equally likely? If not, which is most likely? Least likely? If P(H) < 0.5, then: most likely is TTT; least likely is HHH

The Law of Large Numbers In random sampling, the larger the sample, the closer the proportion of successes in the sample tends to be to the proportion in the population.

The Law of Large Numbers In random sampling, the larger the sample, the closer the proportion of successes in the sample tends to be to the proportion in the population. In other words, the more trials you conduct, the closer you can expect your experimental probability to be to the theoretical probability

The Law of Large Numbers Spinning a Penny

Rolling a Die When you roll a die, (a) what are the outcomes (b) what is the theoretical probability of each outcome?

Rolling a Die When you roll a die, (a) what are the outcomes -- 1, 2, 3, 4, 5, or 6 (b) what is the theoretical probability of each outcome?

Rolling a Die When you roll a die, (a) what are the outcomes -- 1, 2, 3, 4, 5, or 6 (b) what is the theoretical probability of each outcome?

Rolling a Die

A student playing monopoly says “I have not rolled doubles on the last six rolls; I am due for doubles.” How does the Law of Large Numbers apply here?

How does the Law of Large Numbers apply here? A student playing monopoly says “I have not rolled doubles on the last six rolls; I am due for doubles.” The dice will eventually come up doubles, but the probability remains on each roll no matter what has happened before. Any one particular random trial is just that - - random.

Law of Large Numbers In a random process, you can not predict what happens in an individual trial or even in a small number of trials, but you can predict the pattern that will emerge if the process is repeated a large number of times.

Fundamental Principle of Counting Tree diagram: Shows all possible outcomes of an experiment Quickly becomes unwieldy if many stages or many outcomes for stages.

Fundamental Principle of Counting Tree diagram: Shows all possible outcomes of an experiment Quickly becomes unwieldy if many stages or many outcomes for stages. Think about drawing a card from a standard deck of playing cards, replacing it, then repeating this process two more times

Fundamental Principle of Counting If you only need to know how many outcomes are possible, then use the Fundamental Principle of Counting.

Fundamental Principle of Counting For a two-stage process with n1 possible outcomes for stage 1 and n2 possible outcomes for stage 2, the number of total possible outcomes for the two stages is n1 n2. This can be extended to as many stages as desired.

Fundamental Principle of Counting How many outcomes are possible if you flip a coin, roll a die, and pick a card from a standard deck of playing cards?

Fundamental Principle of Counting How many outcomes are possible if you flip a coin, roll a die, and pick a card from a standard deck of playing cards? coin die card 2 ● 6 ● 52 = 624 outcomes

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Fundamental Principle of Counting Suppose you flip a fair coin seven times. a) How many possible outcomes are there? b) What is the probability you will get seven heads? c) What is the probability that you will get heads six times and tails once?

Two-Way Table When a process has only two stages, it is often more convenient to list them using a two-way table.

Two-Way Table When a process has only two stages, it is often more convenient to list them using a two-way table. Make a two-way table that shows all possible outcomes when you roll two fair dice.

Two-Way Table Second Roll 1 2 3 4 5 6 1 First Roll 2 3 4 5 6

Make a two-way table that shows all possible outcomes when you roll two fair dice.

Make a table that gives the probability distribution for the sum of the two dice. The first column should list the possible sums, and the second column should list their probabilities.

Fundamental Principle of Counting Suppose you ask a person to taste a particular brand of strawberry ice cream and evaluate it as good, okay, or poor on flavor and as acceptable or unacceptable on price. How many possible outcomes are there? Show all possible outcomes on a tree diagram. Are all the outcomes equally likely?

Fundamental Principle of Counting 6 outcomes (3 flavor choices times 2 price choices)

Fundamental Principle of Counting Suppose you ask a person to taste a particular brand of strawberry ice cream and evaluate it as good, okay, or poor on flavor and as acceptable or unacceptable on price. How many possible outcomes are there? Show all possible outcomes on a tree diagram. Are all the outcomes equally likely?

Fundamental Principle of Counting b)

Fundamental Principle of Counting Suppose you ask a person to taste a particular brand of strawberry ice cream and evaluate it as good, okay, or poor on flavor and as acceptable or unacceptable on price. How many possible outcomes are there? Show all possible outcomes on a tree diagram. Are all the outcomes equally likely?

Fundamental Principle of Counting

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Questions?