5.1 (Basic concepts in probability)

PROBABILITY RULES The probability of an event is always between ________and ________. In other words, for any event A, _____≤P(A)≤_____ .•If A cannot occur, then _______________.•If A is certain to occur, then_______________.

0 and 1, 0,1 P(A)=0 P(A)=1

EXAMPLE:In a college of 5000 students, 150 are math majors. A student is selected at random and turns out to be a math major. Is this unusual?

150/5000=0.03 yes it is

UNUSUAL EVENTS An unusual eventis one that is not likely to happen. In other words, an event whose probability is small. There are no hard-and-fastrules as to just how small a probability needs to be before an event is considered unusual, but we will use the following rule of thumb.RULE OF THUMB ABOUT UNUSUAL EVENTS:

Any event whose probability is less than 0.05 is considered to be unusual.

the ______________________________consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome.

Empirical method

PROBABILITIES WITH EQUALLY LIKELY OUTCOMES If a sample space has n equally likely outcomes and an event A has k outcomes, then

P (A)= Number of outcomes in A ------------------------------- = k Number of outcomes in the sample space ------- n

In general, if A denotes an event, the probability of event A is denoted by________________.

P(A)

A ______________________________ is one in which we do not know what any individual outcome will be, but we do know how a long series of repetitions will come out. For example, if we toss a fair coin, we do not know what the outcome of a single toss will be,but we do know what the outcome of a long series of tosses will be -about half "heads" and half "tails".

Probability experiment

SAMPLE SPACE The collection of all the possible outcomes of a probability experiment is called a sample space. EXAMPLE:Describe the sample space for each of the following experiments:a) The toss of a coinb) The roll of a diec) The selection of a student at random from a list of 10,000 at a large university

SOLUTION: a)The sample space (heads, tails) b)The sample space (1,2,3,4,5,6) c)The sample space (The 10,000 students)

EXAMPLE:In a recent year, there were 2,046,935 boys and 1,952,451 girls born in the U.S. Approximate the probability that a newborn baby is a boy.

SOLUTION: 2,046,935 2,046,935 + 1,952, 451 ------------------- -------------- = 0.5118 3,999,386 3,999,386

In the Georgia Cash-4 Lottery, a winning number between 0000 and 9999 is chosen at random, with all the possible numbers being equally likely. What is the probability that all four digits are the same?

SOLUTION: There are ten outcomes for all 4 numbers to be the same. And there are 10,000 equally likely outcomes. 10 ------ =0.001 10,000

SAMPLING IS A PROBABILITY EXPERIMENT Sampling an individual from a population is a probability experiment. The population is the sample space and members of the population are equally likely outcomes.EXAMPLE:There 10,000 families in a certain town categorized as follows. A pollster samples a single family from this population.Own a houseOwn a condoRent a houseRent an apartment475314789122857a)What is the probability that the sampled family owns a house?b)What is the probability that the sampled family rents?

SOLUTION:a) 4753/10,000=0.4753 b) 3769/10,000=0.3769

A probability model for a probability experiment consists of a sample space, along with a probability for each event.

The sum of the probabilities is not equal to 1 This is not a probability model.

The ____________________ of an event is the proportion of times that the event occurs in the long run. So, for a "fair" coin, that is, one that is equally likely to come up heads as tails, the probability of heads is 1/2 and the probability of tails is 1/2.

probability

SIMULATION ON THE TI-84PLUS The randIntcommand on the TI-84 PLUS calculator may be used to simulate the rolling of a single die. To access this command, press MATH, scroll to the PRB menu, and select randInt. The following screenshots illustrate how to simulate the rolling of a die 100 times. The outcomes are stored in list L1.

randint (1,6,100), hit sto, 2nd (L1)