Week 2 Probability
Conditional Probability (Example) This table shows the result of a study in which researchers examined a child's IQ and his presence of a specific gene in the child. Find the probability that 1. A child has a high IQ given that the child has the gene. 2. The child does not have the gene. 3. The child does not have the gene given that the child has a normal IQ Gene Present. Gene Absent. Total High IQ. 33 19 52 Low IQ. 39 11 50 Total. 72 30 102
1. Let Event A = Gene Present. Let Event B = High IQ Then P(B|A) = 33/72 2. P (High IQ| Gene Present) P(A) = 72/102 Notice: P (B and A) / P(A) = (32/102) / (72/102) = 32/72 = P(B|A)
Table (Flipping a coin then rolling a 6 sided die)
1. 2. 3. 4. 5. 6 H. H,1. H,2. H,3. H,4. H,5. H,6 T. T,1. T,2. T,3. T,4. T,5. T,6
Example of Independent Events
1. Landing on heads after tossing a coin and rolling a 5 on a single 5-sided die 2. Choosing a marble from a jar and landing on heads after tossing a coin.
Difference between mutually exclusive and independent
1. Two events are different if the occurrence of one of the events does not affect the probability of occurrence of the other event 2. Two events A and B are mutually exclusive if they cannot both occur at the same time
Examples (Mutually Exclusive) Consider: Flip a coin A = heads B = tails
A and B are mutually exclusive but they are not independent because P(B|A) = 0 whereas P(B) = 1/2 (Probability of getting heads and tails on a single coin) Independent Event: Flip a coin twice
Random variables are either discrete or continuous
A random variable is DISCRETE if you can list all the possible outcomes.
Mutually Exclusive Events
Both events cannot occur at the same time
Discrete Probability Distribution
Consists of all the values a random variable can take - Place in the probability distribution function(pdf)
Complement (Tossing a Fair Coin)
Event A: Heads - P (A) = 1/2 - P (B) = P (A^C) = 1/2 - P (A) + P (A^C) = 1
Example ( Dependence / Independence) 1. Selecting a king from a standard deck, not replacing it, then selecting a queen from the same deck. 2. Let Event A be - Selecting a king from a standard deck of cards Let Event B be - Selecting a queen from the same deck Are these events Independent?
Event A: P(A) = 4/52 Event B: P(B) = 4/51 Not Independent (the king was not returned) Dependent because the probability of Event A did affect the probability of event B
What happens when event A and B are not mutually exclusive?
Events A and B will Overlap
Random Experiment
Experiment in which all possible outcomes are known in advance
Probability Distribution Function (pdf)
Experiment: Toss a coin twice and record the number of heads. - The random variable of interest X is the number of heads (Random Variable is UPPER CASE) - The value the random variable X can assume are x=0,1, or 2 ( value is lowercase) P(X=x) is read as the number of heads equals
Finding the Mean of a random variable
For discrete random variables, the symbol for - the expected value or mean is u (mu)- - u = sum of all x values (xp) p = P(X=x) u E(x) = u = summation (xp) 0(0.07) + 1(0.20) + 2(0.38) +3(0.22) + 4(0.13) = 2.14 = Mean
Tree Diagram (Flipping a coin then rolling a 6 sided die)
H 1,2,3,4,5,6 T 1,2,3,4,5,6
List ( Flipping a coin then rolling a 6 sided die)
H:1, H:2, H:3, H:4, H:5, H:6 T:1, T:2, T:3, T:4, T:5, T:6
More Differences
If A and B are mutually exclusive, then if A occurs, B cannot occur, so they are not independent. If A and B are independent, then occurrence of A does not prevent the occurrence of B, so P(B|A) are not mutually exclusive.
Probability
If the number of outcomes is finite (can be counted) and they are all equally likely to occur. - P (A) - Probability that Event A occurs - P (A) - (# Favorable Outcomes/ # Possible Outcomes) - Rolling a 2 - P (A) + P (Rolling a 2) = 1/6
Example (Rule 5) In a group of 20 adults, 4 out of 7 women wear spectacles. Two out of the 13 men wear spectacles. What is the probability that a person chosen at random from a group is a women or someone who wears spectacles?
Let W be event: The person is a woman Let G be event: Person wears spectacles Find P(W or G) = P (W) + P(G) - P( W and G) P(W) = 7/20. P(G) = 4/20 P(W and G) = 4/20 P(W or G) = (7/20) + (6/20) - (4/20) = 9/20
Probability Distribution Function Table Outcome. TT. HT. TH. HH # Heads. 0 1. 1. 2 Probability. 0.25 0.25 0.25. 0.25
Notation: P(X=x) = 0.25 - P(X=1) = 0.25 - P(X=2) = 0.50
Probability Rule #6
P(A and B) = P(B) x P(A)
Probability Rule 5 (Overlapping Events)
P(A or B) = P(A) + P(B) - P(A and B)
Probability Rule 3 (For Mutually Exclusive Events)
P(A or B) = P(A)+P(B)
Probability Rule 4 ( Mutually Exclusive Events)
P(A) + P(A^c) = 1
Definition of Conditional Probability
P(B|A) = P(B and A) / P(A)
Example (Rule 4) Roll a fair 4-sided die
Possible Outcomes: 1,2,3,4 Event A: Rolling a 4 P(A) = P(rolling a 4) =1/4 P(A^c) = P(Rolling a 1,2, or 3) = 3/4
Conditional Probability
Probability of an event occurring given that another event has already occurred - Take event B and also an event A that has already happened - we write this as P(B|A), which is the probability that event B occurs given that Event A has already occurred
Example: Random Experiment
Random Experiment: Rolling a die (fair 6-sided) Sample Space: 1,2,3,4,5,6 Outcome: any of 1-6 Event: Scoring a 6
Rule 3 Example Consider: Fair 6 sided die P(scoring a 1 or a 2)
Scoring a 1 = P(A) Scoring a 2 = P(B) P(A or B) = P(A) + P(B) = 1/6 +1/6 = 1/3
Impossible Event
Scoring a 7 when a six sided die is rolled - The proportion of success is 0 - The probability of any impossible event is zero - Event A is certain - Event A^C is impossible
Sample Space
Set of all these possible outcomes in a random experiment
Event
Subset of the sample space
Example (pdf) A sociologist surveyed the households in Small-town and from this data, constructed the probability distribution function for the number of children in a family. The random variable X represents the number of children in the household Probability Distribution Function x (Lowercase). 0 1 2 3 4 P(X=x). 0.07 0.20 0.38 ? 0.13
Sum of probabilities must equal 1 - 0.07 + 0.20 + 0.38 + ? + 0.13 = 1 - ? = 0.22 - Cant make assumption if not stated that the table is a pdf - What is the probability X is less than or equal to 4? - 1.00
Probability Rule 1
The Probability of an event A, is denoted by P (A) is a number between 0 and 1 - Example: When throwing a die, the probability of getting a 1,2,3,4,5, or 6 is 1/6 - The probability of getting a score of 7 , is zero, since I can never throw a 7
Random Variable
The outcome of interest of a probability experiment - It is a variable because it has more than one possible value - It is random because its actual value is a matter of science
Certainty
The probability of an event that is CERTAIN to occur is 1 - P (1,2,3,4,5,6) = 6/6 = 1
Complement (A^C)
The probability of one event is the COMPLEMENT of another even if the two events do not contain any of the same outcomes AND together they cover the whole sample space
Example (Discrete) Consider: Toss a coin twice Suppose we are interested in the number of tails obtained. Then the random variable is defined as x=number of tails. Since we can count the number of tails obtained, the random variable is discrete
Then the random variable is defined as x=number of tails. Since we can count the number of tails obtained, the random variable is discrete
3 Methods to Identify the sample space of a probability experiment
Tree Diagram, List, Table
Independent Events
Two Events are independent off the occurrence of one of the events does not affect the probability of the occurrence of the other event.
Probability Rule 2
When a fair die is thrown, since all outcomes are equally likely: P (scoring a 1) = P (scoring a 2) = P (scoring a 3) = P (scoring a 4) = P (scoring a 5) = P (scoring a 6) - The sum of all these probabilities is 1
Example: Event
When throwing a die once Event A: Score is 4 = {4} Event B: Score is odd = {1,3,5} Event C: Sore is greater than 5 = {6}
Cumulative Distribution Function (cdf)
x. 0. 1. 2. 3 4 P(X=x). 0.07. 0.27. 0.55. 0.87. 1.00