4.1 Classical Probability

the probability of rolling a 6-sided die and obtaining an even number. There are _____ outcomes in the event

3 - rolling a 2, 4, or 6 - so n(E)=3. There are 66 outcomes in the sample space, n(S)=6, and all events are equally likely. So we calculate the probability to be:

what is the sample space of flipping a fair coin 3x

8 all possible combination of outcomes

Probability of an event:

# of outcomes (meets criteria of "satisfying" event) / total # of outcomes in a sample space

Rounding Rule

Either give the exact fraction or decimal, or round to three digits. If the probability is extremely small, it is permissible to round the decimal to the first nonzero digit. When a probability is not a simple fraction, such as 2/3 or 5/9, express it as a decimal.

Subjective Probability

An educated guess regarding the chance that an event will occur. The accuracy of the probability depends on the expertise of the person giving the probability. Example: A weatherman says there is a 90%90% chance of rain today.

Determine whether the following probabilities is subjective, empirical, or classical. The probability that G32 will be called first in a bingo game.

Assuming that there are 75 numbers in a standard bingo game, all 75 outcomes are known, so this is an example of a classical probability.

Empirical Probability

If E is an event, then P(E), read "the probability that E occurs" is as follows: P(E)=number of times the event E occurs ---------divided by-------- total number of times the experiment is performed=fn Example: A fisherman catches 92 fish in a week at his favorite fishing hole. If 43 of the fish he caught were catfish, what is the probability that the next fish he catches is a catfish? P(catfish)=number of catfish caught / total number of fish caught =4392≈0.4674

outcome

In a given probability experiment, each individual result that is possible. -a result that comes from an experiment

how would you determine the probability of flipping 2 heads in 3 flips of a fair coin

Take the possible outcomes of 2 heads: HHT / HTH / THH and divide by the total possible outcomes in all 3/8 = .375

how would you determine the probability of flipping at least 2 heads in 3 flips of a fair coin

Take the possible outcomes of at least 2 heads: HHT / HTH / THH / HHH and divide by the total possible outcomes in all 4/8 = 2/4 = .5000%

Determine whether the following probabilities is subjective, empirical, or classical. An optometrist wants to know the probability that children will need glasses by the end of 55th grade. She finds that the number of students at the local elementary school (with grades K-5) is 432432 and that 108108 of them have glasses. From this she determines that the probability is 0.25.

The optometrist's probability is based on the statistics from the local elementary school. Since all children were not included, the probability is empirical.

Determine whether the following probabilities is subjective, empirical, or classical. A teacher predicts that 20% of his class will get A's on the final.

The teacher is giving an educated guess, thus a subjective probability.

Subjective Probability can be thought of as

an individual's guess an estimate from experience a judgement

For example:

consider the experiment of closing your eyes and randomly drawing a number out of a hat containing the numbers 1 through 10. In this experiment, there are a total of ten possible outcomes, namely, the numbers 1 through 10. This set of numbers is the sample space. The event "draw an even number" is just the specific set of outcomes 2, 4, 6, 8, 10. If any of these specific outcomes were to occur, we would say "an even number was drawn".

classical probability

most precise type of probability since it is calculated by taking all possible outcomes for an experiment into account. "the probability that E occurs" is as follows: where n(E)= the number of outcomes of an event, and n(S)= the number of outcomes in the sample space

Empirical Probability can be thought of as

no assumptions evidence supports it based on what is seen in an experiment roll 100x and if you roll 15 threes the chance of probability is: 15%

how would you determine the probability of drawing a spade

outcomes: 13 (there are 13 spades in a deck) sample space: 52 (there are 52 total cards) probability: 13/52 = .25%

how would you determine the probability of drawing a two or a Queen

outcomes: 8 (there are 4-2's & 4-Q's in a deck) sample space: 52 (there are 52 total cards) probability: 8/52 = 2/13 or .1539%

probability experiment

process that produces outcomes or trial, is any process in which the result is random in nature. Examples of probability experiments include flipping a coin, tossing a pair of dice, or drawing a raffle ticket. there is more than one possible result and the result is determined at random

The set of all possible outcomes for a given probability experiment is called the

sample space a.k.a. probability space all mutually exclusive outcomes of an expirament

event

set of 1 or more possible outcomes -is a subset of outcomes of the sample space.

what is the outcome of flipping a fair coin 1x

heads or tails

Law of Large Numbers

the greater the number of trials, the closer the empirical probability will become to the true probability

tree diagram

to be used when an experiment consists of several stages, it allows the outcomes to be organized in a systematic manner. The tree begins with the possible outcomes for the first stage and then branches for each additional possibility.

When tossing a coin, there are _____ in the sample space

two outcomes heads and tails

thought expirament

you don't even need an actual dice - not a test, not from a trial "abstract"