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Probability

Basic Rules of Probability

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Definition:

P denotes a probability.
A, B and C are specific events.
P(A) is the probability that an event A will occur.

Rule for computing probability of equal likely events

$$P(A) = \frac{\text{the number of ways {\it A} can occur}}{\text{number of different simple events}} = \frac{s}{n}$$

Example:

Suppose you toss 3 coins. What is the probability of getting exactly 2 heads? Solution: Let A be the event for rolling exactly 2 heads. To compute P(A) we need to know the total number of combinations of rolling three dice. There are 8 possible outcomes for the three coins. Of those 8 combinations there are 3 ways to get exactly two heads. So,

$$P(A) = \frac{3}{8} = .375$$

Definition:

A compound event is any event combining any two simple events.

The notation

P(A or B) = P(event A or event B occurs or they both occur)

Rule:

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

where P(A and B) denotes the probability that A and B both occur at the same time.

Definition:

Events A and B are mutually exclusive if they cannot happen simultaneously.

Definition:

If A is an event then the compliment, A, consists of all the outcomes in which event A does NOT occur.

Rule for compliments

If A is an event then,

P( A) = 1 - P(A).

Example

A six sided die is tossed. What is the prabability of NOT rolling a one? Solution: Let A be the event of NOT rolling a one. Then A is the event of rolling a one. Since P( A) = (1/6), the probabilty of not roling a one is (5/6) (= 1 - P( A)).

Conditional Probability

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Definition:

The conditional probability of an event B after it is assumed that the event A has already occurred is denoted by P(B|A).

Example:

Suppose you have 3 green dice and 2 red dice. You pick a die at random and role it. Let A be the event the die is green. Let B be the event that the top number is even and the die is green. What is P(B)? What is P(B|A)? Solution: To compute P(B) we notice that there are 5 dice with 6 faces each, for a total of 30 possible equal likely outcomes. There are 3 green dice with 3 even numbers each, giving a total of 9 ways B can occur. Hence, P(B)=9/30=.3.

To find P(B|A) we can assume that the die that was rolled is green. The probability of getting an even role is (1/2), since there are 3 even numbers on the die out of 6 possible choices.

Definition:

Two events A and B are independent if the occurrence of one does NOT affect the probability of the occurrence of the other. If A and B are not independent, they are said to be dependent.

Rule:

Given events A and B, P(A and B) = P(A) • P(B|A) and

P(B|A) $=\dfrac{P(A and B)}{P(A)}$

Test for Independence

Two events A and B are independent if	Two events A and B are dependent if
P(B|A)=P(B)	P(B|A) ≠P(B)
or	or
P(A and B) = P(A) • P(B)	P(A and B) ≠P(A) • P(B)

Counting(optional)

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Counting Rule:

If the event A can occur m ways and the event B can occur n ways then events together can occur in m • n ways

Example:

Suppose that a bank issues you a personal identification number (PIN). The PIN consists of 3 letters followed by a digit. How many different possible PIN's are there? Solution: Since there are 26 letters and 10 digits, there are 26 • 26 • 26 • 10 =175,760 possibilities.

Another counting Rule:

The number of sequences (the order in which the items are selected matters here) of r items selected from n available items is

$${}_nP_r = \frac{n!}{(n-r)!}.$$

The number of groups (the order in which the items are selected does NOT matter here) of r items selected from n available items is

$${}_nC_r =\frac{n!}{(n-r)!r!}.$$

Where $n! = n\cdot(n-1)\cdot(n-2)\cdot\ldots\cdot 2\cdot 1$.

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