Probability P{E} Example Consider. Find P{HH}. simultaneously. = # ways E occurs

Transcription

1 Probability and the Binomial Distribution Definition: A probability is the chance of some event, E, occurring in a specified manner. NOTATION: P{E} We can view probabilitie es from a Relative Frequency Interpretation Example Consider the experiment where a fair coin is tossed twice. P{E} = # ways E occurs # total events Find P{HH}. We call the collection of all possible events a sample space and we denotee the sample space by. Definition Two events, E 1 and E 2 are disjoint (or mutually exclusive) if they cannot occur simultaneously Definition The union of two events, E 1 and E 2 is the event that E 1 or E 2 or both occur NOTATION: E1 E 2 E 1 or E 2 Definition The intersection of two events, E 1 and E 2 is the event that E 1 and E 2 both occur NOTATION: E1 E 2 E1 and E 2 Definition Two events, E 1 and E 2 are independent if knowledge that E 1 occurred does not affect P{E 2 } and vice versa. Two events that are not independent are dependent. Definition A conditiona l probability is the probability that one event occurs given that anotherr event occurred. NOTATION: E1 E 2 (reads E 1 given E 2 ) A common way to visualize how events interrelate in a sample space is through a Venn Diagram.

2 Probability Rules Rule 1 0 P{E} 1 Rule 2 If E 1, E 2,, E k are all the possible (disjoint) events, P{E i } = 1 Rule 3 If E c = {not E}, then P{E c } = 1 P{E} Rule 4 P{E 1 E 2 } = P{E 1 } + P{E 2 } P{E 1 E 2 } Rule 5 If E 1 and E 2 are disjoint, then P{E 1 E 2 } = P{E 1 } + P{E 2 } Rule 6 P{E 1 E 2 } = P{E 1 E 2 }P{E 2 } = P{E 2 E 1 }P{E 1 } Rule 7 If E 1 and E 2 are independent, then P{E 1 E 2 } = P{E 1 }P{E 2 } Chapter 3 Page 2

3 Examples Computing Probabilities Example The following table summarizes the outcome of a study recording hair color and eye color for 1770 German men. For a randomly chosen male from this group, find P{Black Hair} P{Black or Red Hair} P{Black Hair or Blue Eyes} P{Red Hair and Brown Eyes} P{Blue Eyes Black Hair} Chapter 3 Page 3

4 Tree Diagram A tree diagram is a useful tool for computing probabilities. Each branch in the tree diagram represents possible outcomes for a piece of an experiment. Chapter 3 Page 4

5 Example The ELISA (Enzyme Linked ImmunoSorbent Assay) is a common way to test for HIV infection. According to the CDC (the web page has since been unlinked) for the ELISA test, we have sensitivity = P{true positive} = P{test positive you have the disease} 99.9% specificity = P{true negative} = P{testing negative you don t have the disease} 99.8% positive predictive value (PPV) = P{you have the disease test positive} For a certain population, where 1/1000 are HIV positive, use a tree diagram to find: P{have HIV test positive} and P{have HIV test negative} Chapter 3 Page 5

6 As was mentioned before in class, a random variable iss the measured outcome of some random process. When the random variable is quantitative, we can either have a discrete or a continuous random variable. When we have a discrete distribution of a random variable, we can list the probability associated with each possible outcome. From examplee 3.5.5, consider a certain population of the freshwater sculpin, Cottus rotheus. The distribution of the random variable Y = number of tail vertebrae is shown in the following table We ve listed out the entire probability distribution all the probabilities add up to one. We could graphically represent a discrete distribution with a frequency histogram. Construct a frequency histogram for the number of tail vertebraee in this population of sculpin. Chapter 3 Page 6

7 In the case where we have a continuous random variable, we want a different tool to represent this type of distribution. We call this representation of a continuous random variable s distribution a density curve. Consider the distribution of blood glucose levels measured one hour after a subject (from a certain population off women) drinks 50mg of glucose dissolved in water. Example depicts this distribution with binwidth set 10, 5 and 0 respectively Notice the probability density curve is like having a probability histogram where we re squeezing the binwidth down to 0 (an infinite number of bins). Then, the way we get probabilities associated with continuous random variables is still an area, just like in the frequency histogram. But, we need area under a curve, so we need calculus. Integration from a to b of the function that results in the probability density curve will give us the probability off being between those two values under the specified distribution. Fortunately, your TI calculator or a computer will be doing the integration for you in this class! More on that later Now, think back to your math classes. Questions: What is the length of a single point? What is the area of a line? The following facts pertaining to probabilitie es associated with a continuous random variable are consequences of the fact that a line doesn t have area: P{Y = a} = 0 = P{Y = b} (area of a line is zero) So, P{Y a} = P{Y < a} + P{Y = a} = P{Y < a} And, P{a Y b} = P{a < Y b} = P{a Y < b} = P{a < Y < b} Chapter 3 Page 7

8 Mean and Variance of a Discrete Random Variable The population mean of a discretee random variable, Y,, is given by Y = y i P{Y = y i } The mean of a random variable, Y, is also known as thee expected value of Y, denotedd E[Y]. The population variance of a discrete random variable, Y, is given by 2 Y = (y i Y ) 2 P{Y = y i } One can show that 2 Y = E[Y 2 ] (E[Y]) 2 = E[Y 2 ] ( Y ) 2 2 Then, the population standard deviation is Y = σ Y Example For the number of tail vertebrae in the population of freshwater sculpin, Cottuss rotheus, find Y and Y. Chapter 3 Page 8

9 Let s refresh our memory on computing probability with an example. Consider the experiment where we toss a fair coin 3 times. Find the probability distribution for flipping heads in this experiment. Now, think about finding probability distributions associated with flipping a fair coin say 6 times. And then consider the experiment where the coin is not fair. The calculations get unwieldy fast! We need a more convenient method Chapter 3 Page 9

10 Definition: The independent trials model occurs when (i) n independent trials are studied (ii) each trial results in a single binary observation (iii) each trial s success has (constant) probability: P{success} = p Notice that if P{success} = p, P{failure} = 1 p. Your text calls this the BInS (Binary / Indep. / n is fixed / Same p) setting, but is commonly referred to as a Binomial Experiment In a BInS setting, if we let Y = {# successes} then Y has a binomial distribution. NOTATION: Y ~ Bin(n,p). The binomial probability function is P{Y = j} = n C j p j (1 p) n j j = 0,1,,n where n C j = n! j! n j! with j! = j(j 1)(j 2) (2)(1) and define 0! = 1 Example Use the binomial probability function to find P{exactly 1 head} in the experiment where a fair coin is flipped 3 times. Find P{at least one head} Chapter 3 Page 10

11 The TI calculators will compute binomial probabilities. For P{Y = j} Choose 2 nd and VARS to bring up DISTR menu > scroll down to binompdf > ENTER > binompdf(n,p, j) For P{Y j} choose 2 nd and VARS to bring up DISTR menu > scroll down to binomcdf > ENTER > binomcdf(n,p,j) From Example Suppose thatt 37% of the individuals in a large population have a certain mutant trait and that a random sample of 5 individualss is chosen from the population. Find P{at least 1 and at most 4 mutants} in the sample The following is a probability histogram of the distribution from example Find the mean of the distribution of the number of mutants. Binomial mean and variance If Y ~ Bin(n,p),, the population mean and variance reduce to: µ Y = np and Y 2 = np(1 p) Example Find the mean and standard deviation for thee number of mutants out of five selectedd individuals from the population described in example Chapter 3 Page 111