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TERMS IN PROBABILITY

Experiment: is a process that produces a single outcome whose result cannot be predicted with certainty
Sample Space: a collection of all outcomes that can result from a decision, selection, or experiment.
Sample Point: each individual outcome in the sample space. its also referred to as “element”
Discrete sample space: it contains finite or infinite countable number of elements.
Continuous sample space: consist of points on the line segment, (decimals)
Events: set of favorable outcomes in a sample space.

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TYPES OF EVENTS

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INDEPENDENT EVENTS

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DEPENDENT VARIABLE

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EXAMPLES

In a certain test 5 out of 20 students scored an ‘A’. We chose three student at random out of the 20 students without replacement. Find the probability that all three are the ones who scored an ‘A’.

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BASIC RULES OF PROBABILITY

RULE 1: probability of a random experiment lies within 0 and 1

0 ≤ P(E) ≤ 1

RULE 2: Probabilities of all possible outcomes is 1.

∑ P(E ) = 1

RULE 3: The probability of an event E is equal to the sum of the probabilities of the individual outcomes forming E.

P(E )= P(e1)+P(e2)+P(e3)

RULE 4: The probability of any event E1 or E2 is equal to the sum of the individual probabilities of the events outcomes forming E1 and E2 minus the probability of events E1 and E2 occurring.

P(E1 or E2) = P(E1) + P(E2) – P(E1 and E2).

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Examples

In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?

A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a king or a club?

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RULE 5: the probability of mutually exclusive events is equal to the sum of the probabilities of the individual events

P (E1 or E2) = P(E1) + P(E2)

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FINDING THE EXPECTED VALUE AND STANDARD DEVIATION OF PROBABILITY DISTRIBUTION

The expected value of a probability distribution is the mean of that distribution.
Mathematically, the expected value is given by:

E(x) = ∑ xP(x)

where E(x) – expected value of x

x- values of the discrete random

P(x) – probability of the random variable taking on the value x.

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Size of household	1	2	3	4	5	6	7 or more
Probability	26.7%	33.6%	15.8%	13.7%	6.3%	2.4%	1.5%

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Examples

Suppose a there is a 20% chance of 1 inch of rain, a 70% chance of 2 inches of rain, and a 10% chance of 3 inches.
Suppose in a certain game there is a 5% chance of winning $100, a 50% chance of winning $0 and 45% chance of losing $20.
Find the standard deviation of a probability distribution

Goals (x)	Probability P(x)
0	0.18
1	0.34
2	0.35
3	0.11
4	0.02

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Examples

The following probability distribution tells us the probability that a given vehicle experiences a certain number of battery failures during a 10 year span

what is the standard deviation of the number of failures for each vehicle.

Failures (x)	Probability P(x)
0	0.24
1	0.57
2	0.16
3	0.03

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ASSIGNMENT

A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. What is the expected value and standard deviation?

x	P(x)
0
1
2
3
4
5

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ASSIGNMENT

The probability distribution tells us the probability that a certain soccer team scores a certain number of goals in a given game:
Find the expected value and standard deviation?

Goals(X)	Probability P(x)
0	0.18
1	0.34
2	0.35
3	0.11
4	0.02