PROBABILITY THEORY

RULES OF PROBABILITY - review

Rule 1: 0 P(A) 1 for any event A– Any probability is a number between 0 and 1

Rule 2: P(S) = 1– All possible outcomes together must have a

probability of 1

– Some outcome must occur on any trial

Rule 3: For any event A,

P(A does not occur) = 1 – P(A)

Rule 4: Addition rule: If A and B are disjoint events, then

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

• Example– All human blood can be typed as one of O, A,

B, or AB, but the distribution of the types varies a bit with race. Here is the probability model for the blood type of a randomly chosen black American:

Blood type O A B ABProbability 0.49 0.27 0.20 ?

A: What is the sample space of blood type for black Americans?

B: What is the probability of AB type? Why?C: What is the probability of type O or A?D: What is the probability that an individual is not

type O?E: What is the probability that an individual has the

substance A in his/her blood?F: Maria has type B blood. She can safely receive

blood transfusions from people with blood types O and B. What is the probability that a randomly chosen black American can donate blood to Maria?

PROBABILITY THEORY

• When the outcome of one event does not affect or predetermine the outcome of another event, we say that the two events are INDEPENDENT.

EXAMPLE 2:• A standard 52-deck of cards contains 26

red, and 26 black cards. For the first card dealt from a shuffled deck, the probability of a red card is _______. Given that a red card is drawn and removed from the deck, what is the probability of drawing a red card on second trial?

• Are these events independent?

A: draw a red card on first try

B: draw a red card on second trial

PROBABILITY THEORY

VENN DIAGRAM• Diagram that displays a sample space and

events within it• Depicts relationships among events

Fig 5.1Fig 5.2

PROBABILITY THEORY

MULTIPLICATION RULE FOR INDEPENDENT

EVENTS

• Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. For two such events

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

PROBABILITY THEORY

EXAMPLE: Select a first-year college student at random and ask what his or her academic rank was in high school. Here are the probabilities based on proportions from a large sample survey of first-year students:

• A: Choose two first-year college students at random. Why is it reasonable to assume that their high school ranks are independent.

• B: What is the probability that both were in the top 20% of their high school classes?

• C: What is the probability that the first in the top 20% and the second was in the lowest 20%?

Rank Top 20%

Second 20%

Third 20%

Fourth 20%

Lowest 20%

Probability

0.41 0.23 0.29 0.06 0.01

Probability

• Intervals of outcomes

– P(0.4 ≤ X ≤ 0.8) =_____________

– P(X ≤ 0.5) = _________________

– P(X > 0.8) = _________________

– P(X ≤ 0.5 or X > 0.8) =_________

A = 1

1

10

ProbabilityNormal Probability Distributions

– Any density curve can be used to assign probabilities

– Normal distribution – probability model

– Heights of all young women follow a normal distribution with μ of 65.5 inches and σ if 2.5 inches.

– This is a distribution for a large set of data.

– If you choose any woman C at random, and repeat the randomization many times, the distribution of values of C follow normal distribution.

Normal Probability Distributions

Example:– What is the probability that a

randomly chosen young woman had height between 68 and 70 inches?

Distribution of Height =N=(64.5, 2.5)

Remember – Z table finds probabilities of standardized data.

5.2

5.6470

5.2

5.64

5.2

5.6468)7068(

XPXP

0669.0

9192.09861.0

)2.24.1(

ZP

Probability

Example– The random variable X has the

standard normal N(0,1) distribution. Find each of the following probabilities:

A: P(-1 ≤ X ≤ 1)

B: P(1 ≤ X ≤ 2)

C: P(0 ≤ X ≤ 2)