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
