Topic 4: Probaility
What is the probability of an event that will ALWAYS occur?
1.0 is the probability of an event that will ALWAYS occur.
According to a study published by the National Institute of Health, during intercourse condom breakage rates fall between 0.4% and 2.3%. Suppose that the breakage rate of Brand X is 2%. A Brand X Pleasure Pack contains 12 condoms. If L is the event that at least one of the 12 condoms breaks during intercourse, what does P(not L) represent (in words)?
None of the condoms break
two fundamental ways in which we can determine probability:
Theoretical (also known as Classical) Empirical (also known as Observational)
A coin is tossed three times, or until the first "head" appears. (whichever occurs first):
We stop the experiment when we get the first "head" (H, TH, TTH), or after three tosses, even if we didn't get any "heads" (TTT).
Sample Space (Probability)
the set of all possible outcomes of an experiment.
Given that
to express the probability of a certain event given that another event holds. Notice how this new notation if formed: The event whose probability we seek (in this case E) is written first. The vertical line stands for the word "given" or "conditioned on." The event that is given (in this case M) is written after the "|" sign: P(E|M)
Law of Large Numbers
(statistics) law stating that a large number of items taken at random from a population will (on the average) have the population statistics
Let A and B be two disjoint events such that P(A) = .20 and P(B) = .60. What is P(A and B)?
0 If two events are disjoint, then by definition, the two events cannot happen together. The probability of these two events happening together is denoted by P(A and B). If this is impossible, then P(A and B) = 0.
Which of the following is the probability of an event that will NEVER occur?
0.00 is a probability of an event that will never occur.
Which of the following represents the probability of an event that IS more likely NOT to occur, but its occurrence is not very rare.
0.25 Since P(A) = 0.25 is less than 0.50, event A will not occur more often than it will occur. Since event A will occur 25% of the time, its occurrence is not very rare.
According to a study published by the National Institute of Health, during intercourse condom breakage rates fall between 0.4% and 2.3%. Suppose that the breakage rate of Brand X is 2%. A Brand X Pleasure Pack contains 12 condoms. If L is the event that at least one of the 12 condoms breaks during intercourse, what does P(not L) ?
0.98^12 P(none of the condoms break)= P(1st does not break and 2nd does not break and ..... and 12th does not break)= * Using our assumption of independence: * = (0.98)(0.98)(0.98)......(0.98)=0.98
Rule 1
0<_ P(A) _<1 it can range anywhere form 0 (the event will never occur) to 1 (the event is certain)
According to a study published by the National Institute of Health, during intercourse condom breakage rates fall between 0.4% and 2.3%. Suppose that the breakage rate of Brand X is 2%. A Brand X Pleasure Pack contains 12 condoms. What is the probability that at least one of the condoms in a Brand X Pleasure Pack will break during intercourse? In other words, what is P(L)?
1-(.98^12) P(L)=1-P(not L)= 1-P(none of the condoms break)= 1-(.98^12). Note that 1-(.98^12)=0.215, and so there is more than 20% chance that at least one of the condoms in the package breaks
Which of the following is NOT a legitimate probability of an event?
1.001 is not a legitimate probability of an event since probability cannot be a number that is larger than 1.
Flipping a Fair Coin
A coin has two sides; we usually call them "heads" and "tails." A coin that is not unevenly weighted and does not have identical images on both sides is called a "fair" coin. For a "fair" coin the chances that a "flip" will result in either side facing up are equally likely. Thus, P(heads) = P(tails) = 1/2 or 0.5. Letting H represent "heads," we can abbreviate the probability: P(H) = 0.5. Classical probability
This event is impossible
A probability of 0 represents an event that can never occur.
This event is extremely unlikely, but it will occur once in a while in a long sequence of trials
A probability of 0.01 represents an event that is VERY unlikely, but is still possible.
This event will occur more often than not, but is not extremely likely
A probability of 0.60 represents an event that will occur more often than not. Specifically, it will occur almost two-thirds (0.67) of the time.
P(A or B) and of P(A and B)
Addition Rule, we can add their probabilities (i.e., replace every "or" with +). Also, within each of the four possibilities, we can use the Multiplication Rule and replace "and" with *
A taco truck sells 5 varieties of tacos. Assume Event V occurs when a customer purchases a vegan taco. Which of the following statements would indicate that most customers do not buy vegan tacos, however it is not an extremely rare event?
An event will never occur if it has a probability of 0.00. P(V) = 0.12 indicates that this event will not happen most of the time. However, it is not an extremely rare event for a customer to purchase a vegan taco.
Theoretical (also known as Classical)
Classical methods are used for games of chance, such as flipping coins, rolling dice, spinning spinners, roulette wheels, or lotteries. They are "classical" because their values are determined by the game itself.
What is the term for the set of all possible outcomes in an experiment or random trial?
Defining the list of all possible outcomes in the sample space is an excellent first step in figuring out the probabilities associated with the events.
Rolling Fair Dice
Each traditional (cube-shaped) die has six sides, marked in dots with the numbers 1 through 6. On a "fair" die, these numbers are equally likely to end up face-up when the die is rolled. Thus, P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6 or about 0.167. Classical probability
Suppose that P(A) = 0.65. Which of the following best describes event A?
Event A will occur more often than not, but it would not be unusual for event A not to occur.
Rule 5
P(A and B) = P(event A occurs and event B occurs)
General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B) where, by P(A or B) we mean P(A occurs or B occurs or both). In the special case when A and B are disjoint events (which means that P(A and B)=0), the general addition rule becomes P(A or B)=P(A)+P(B). This rule is called the Addition Rule for Disjoint Events.
Rule 4 disjoint events
P(A or B) = P(A) + P(B).
Rule 2
P(S)=1; the sum of the probabilities of all possible outcomes is 1
Rule 3: The Complement Rule
P(not A) = 1 - P(A); that is, the probability that an event does not occur is 1 minus the probability that it does occur. P(A)=1-P(not A)
A quiz consists of 10 multiple-choice questions, each with four possible answers, only one of which is correct. A student who does not attend lectures on a regular basis has no clue what the answers are, and therefore uses an independent random guess to answer each of the 10 questions. What is the probability that the student gets at least one question right?
P(question right)=P(R)=1/4=.25 P(the question wrong)=P(W)=3/4=.75 P(not L)=P(getting all the questions wrong)= P(W1 and W2 and W3 and W4 and W5 and W6 and W7and W8 and W9 and W10) P(not L)= .75*.75*.75*.75*.75*.75*.75*.75*.75*.75=.0563 P(L)=1-P(not L)=1-.0563=.9437
A fair traditional 6-sided die is rolled 100 times and the number 5 appears 20 times. What is the empirical probability of rolling a 5 in this situation?
Since in 20 of the 100 rolls the die showed 5, the empirical probability of getting a 5 is 20/100 =0.20 or 20%. Nice work!
A fair traditional 6-sided die is rolled once. What is the probability of rolling a 5?
Since there are 6 sides on the dice you will have a 1/6 (or approximately .17) chance to roll a five if you roll the dice once.
Notation
The act of noting, marking, or setting down in writing. If we wish to indicate "the probability it will rain tomorrow," we use the notation "P(rain tomorrow)." We can abbreviate the probability of anything. If we let A represent what we wish to find the probability of, then P(A) would represent that probability. We can think of "A" as an "event."
Empirical (also known as Observational)
The empirical way for finding probability uses a series of trials to determine (actually, estimate) the probability of an event. Each such trial produces outcomes that cannot be predicted in advance. empirical probability determined based on experience. The empirical way to determine probability is experiential or observational.
A fair traditional 6-sided die is rolled 100 times and the number 5 appears 20 times. Which of the following it true regarding the theoretical (classical) probability of getting a 5 and the empirical probability of getting a 5? (If necessary, round your answers to 2 decimal places.)
The theoretical probability is 0.17 and the empirical probability is 0.20. Since the die is fair, the theoretical probability of getting a 5 is 1/6 or approximately 0.17. Since in 20 of the 100 rolls the die showed 5, the empirical probability of getting a 5 is 20/100 =0.20.
Spinners
This particular spinner has three colors. However, each color is not equally likely to be the result of a spin, since the portions are not the same size. Since the blue is half of the spinner, P(blue) = 1/2. The red and yellow make up the other half of the spinner and are the same size. Thus, P(red) = P(yellow) = 1/4. Classical probability
P(S) represents the likelihood that a customer chooses extra hot habanero sauce over salsa on their taco. P(S) = 0.03 indicates that customers rarely order extra hot habanero sauce.
True- a probability of 0.03 indicates an event rarely occurs.
At a party with 60 guests, the likelihood that at least 2 guests share the same birthday is
Very High. There is over a 99% chance of 2 guests having the same birthday. This demonstrates that probability is not always intuitive.
A pair of dice is rolled, and the sum of the dots on the two faces that come up is recorded:
When rolling two dice, the possible outcomes are: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) ... (6,6). Thus, the sums can be 2-12.
probability sample
a sample in which each member of the population has some known chance of being included. We use probability to quantify how much we expect random samples to vary. This gives us a way to draw conclusions about the population in the face of the uncertainty that is generated by the use of a random sample. One way to think of probability is that it is the likelihood that something will occur.
Two events that cannot occur at the same time are called
disjoint or mutually exclusive.
In probability, "OR" means
either one or the other or both. (+) Larger probability.
relative frequency
the fraction or percent of the time that an event occurs in an experiment. If we toss a coin, roll a die, or spin a spinner many times, we hardly ever achieve the exact theoretical probabilities that we know we should get. But we can get pretty close. When we run a simulation or when we use a random sample and record the results, we are using empirical probability. This is often called the Relative Frequency definition of probability.
Conditional Probability
the probability of an event ( A ), given that another ( B ) has already occurred.
In probability, "AND" means
will always be associated with the operation of multiplication. Smaller probability.
