A patient is said to be hypokalemic (low potassium in the blood) if the measured level of potassium is 3

Math 37 - Lecture 11

Probability
A patient is said to be hypokalemic (low potassium in the blood) if the measured level of potassium is 3.5 meg/l or less. You have tested hypokalemic. Do you think you really are?

Example An individual's potassium level varies daily, and there is some variation in the measurement procedure. Suppose the overall variation follows a Normal distribution. If you have a mean potassium level of 3.8 with a standard deviation of .2, and you are measured several times, according to this model, on what proportion of days will the measurement indicate hypokalemia?

If you are not hypokalemic, what is the probability the test indicates you are?

Def: We call a phenomena random if individual outcomes are uncertain, but there is a regular distribution of outcomes in a large number of repetitions. The probability of an outcome is the proportion of times the outcomes would occur in a very long series of repetitions.

Def 1: The probability of an outcome is the likelihood of it occurring.

Def 2: Suppose we could repeatedly record an outcome forever. The probability of the outcome is the long term relative frequency.

Examples: Repeatedly tossing a coin, seeing how many land heads.

- Toss a tack, how often does it land point down?

A Probability Model describes the randomness we see in a phenomena

1. List all the possible outcomes = sample space

2. Assign a probability to each outcome: long-run relative frequency

(# observations/# tries) or personal probability or assume equally likely

Basic Probability Rules (for assigning probabilities to outcomes)

1. Any probability is a number between 0 and 1

2. All possible outcomes together must have probability 1

Examples Probability newborn is male, Roll two dice

Example Consider the following five outcomes for an experiment in which the type of ice cream purchases by the next customer at a certain store is:

Brand	Steve's Pint	Ben and Jerry's Pint	Dreyer's Quart	Dreyer's Half gallon	Von's Half gallon
Probability	.10	.15	.20	.25

- What is the sample space of this phenomena?

- For this to be a legitimate probability model, what is the probability the next customer purchases Von's Half Gallon?

- What can you tell me about the probability that the size purchased is larger than a pint?

Def: An event is a set of outcomes (subset of sample space). Assign probabilities by summing the probabilities of the outcomes in the event.

Special Case: Equally Likely Outcomes

If have equally likely outcomes, and A in an event then P(A)=

Examples - Probability of 3 heads in 4 coin tosses

What is the probability that the ice cream size is a pint or a half gallon?

Def: The Addition Rule allows you to combine events when both events cannot occur simultaneously (disjoint, mutually exclusive)

P(A or B) = P(A) + P(B)

The probability 2 successive customers both purchase pints?

Def: The Multiplication Rule allows you to combine events when the events are independent (the chance of one outcome occurring is not affected by knowledge of whether or not the other occurred.

P(A and B) = P(A)P(B)

What's the probability that neither purchases pints?

Def: The Complement of an Event is the set of outcomes in the sample space that are not in the event. P(A^c) = 1- P(A)

Misconceptions and Cautions

1. Law of Averages/Gambler’s Fallacy

"Just tossed 10 straight heads, I must be due to get some tails"

Independent events have no memory!
2. Myth of Short-run Regularity (Law of Small Numbers)

"He just made 10 shots in a row, he has the touch tonight!"

Regularity and Randomness is in the Long run!

Example Good ole Mars M&Ms

Color	Brown	Red	Yellow	Green	Orange	Blue
Probability	.3	.2	.2.	.1	.1

Find the probability you select (a) Brown or Red (b) Christmas or UOP Color (c) Not yellow (d) Not yellow nor tan (e) A brown the first time and then a red the second time assumptions?

Example At a streetlight, the probability you hit a red light is .4, yellow is .2 and green is .4. What is the probability you hit green or yellow? What is the probability you hit 2 green in a row?