Probabilities

when all objects being permutated are identical

(all counted letters)/ each letter and how its reapeated

0!

1

How many ways can a person toss a coin 11 times so that the number of heads is between 7 and 10 inclusive?

11C7+11C8+11C9+11C10

multistage experiment

2 step experiment

A standard pair of six-sided dice is rolled. What is the probability of rolling a sum of 12?

36possible pairs of numbers.

A standard pair of six-sided dice is rolled. What is the probability of rolling a sum less than 6?

5/18

complement for E

Ec(exponent), consists of all the outcomes in the sample space that are not in E

What is the probability of rolling a number greater than or equal to 4? Express your answer as a simplified fraction or a decimal rounded to four decimal places.

Find all possibilities for each of the sum

experimental

P(E)=f/n f= frequency of an event n=total number of times the experiment is performed.

A coin is tossed 5 times. What is the probability of getting all tails?

So n=1. Moreover, there are a total of (2)5 fifth power =32 different outcomes possible when tossing a coin 5 times. So n(S)=32.

event

Subset outcome from the sample space

How many different account numbers are possible if repetitions of letters and digits are not allowed?

We use the multiplication of each slot. For example is the combination is 2 letters and 4 numbers than the way we can find a solution will be: 26*26*10*10*10*10

Probability Experiment

any process with a result determined by chance

probability is always

between 0 and 1

key terms

define the method that must be used in order to obtain the correct answer

Outcome

each individual result

Subjectivity

educated guess regarding the chance that an event will occur

permutation

is a selection of elements chosen from a group where the arrangement is specific

r

objects are selected from a group of n distinct objects

mutually exclusive

share no outcomes

3 methods for calculating probability

subjectivity, experimental and classical

Law of Large Numbers

the greater the number of trials, the closer the experimental probability will be to the true probability.

Classic probability

the most practice probability. Calculates ALL possible outcomes and can only be applied when all outcomes are equally likely. P(E)= E/S. E is the number of outcomes in the event and S is the number of outcomes in the sample space

factorial

the product of all positive integers less than or equal to a given positive integer, n. equal the products of strings of positive numbers

Sample Space

the set of all possible outcomes

conditional probability

two events happening consecutively in a multistage experiment when the events are not independent.

How many different account numbers are possible if repetitions of letters and digits are not allowed?

when not allowed we use the permutation

If a coin is tossed 3 times, and then a standard six-sided die is rolled 3 times, and finally a group of three cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?

2(3rd power)* 3(3rd power)* 52C3

tree diagram

Allows patterns to be be organized sytematically

combination

is a selection of elements chosen from a group without regard to their order.

Choosing a diamond or a black card out of a standard deck of cards.

mutually exclusive