basic terms of probability dalesandro

7/29/2019 Basic Terms of Probability Dalesandro

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Basic Terms of Probability

Section 3.2


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Definitions

Experiment:A process by which an

observation or outcome is obtained.

Sample Space: The set S of all possible

outcomes of an experiment.

Event: Any subset E of the sample space

S.


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Probability of an Event

Probability of an event is a measure of the

likelihood that the event will occur.

Remember probability is a number not a set.

Mathematically speaking the probability of an

event E denoted by P(E) is:

P(E) = n(E)/n(S).

Recall that that n(E) is the cardinal number of

set E and n(S) is the cardinal number of set

S.


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Odds of an event

Dont confuse probability with odds.

Every state lottery supposedly tells you the odds ofwinning.

It turns out they dont. They tell you the probability of

winning. By definition, the odds of event E happening are denotedby o(E), which is given by:

o(E) = n(E):n(E).

In words this says, that the odds of event E happening

are number of times E happens divided by the number oftimes E does not happen.

Think success compared with failure.

Note odds use : , think of this as a fraction symbol ordivision sign.


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NJ state Lottery

According to the New Jersey state lottery, the odds ofwinning the Pick 6 Jackpot is 1:13,983,816.

However this is not correct. What they tell you is theprobability.

Remember the number of Pick 6 numbers is 49C6 . Thisnumber equals 13,983,816.

The sample space contains 13,983,816 numbers and thewinning number is just one of these. Hence a probability.

The correct way to state the odds would be1:13,983,815.

There is 1 successful number and 13,983,815unsuccessful number.


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Relative frequency

Tossing a single coin has a sample space of Heads and Tails.That is S={H,T}.

Theoretically speaking the probability of tossing a head is .

Let say you flip a coin 10 times and record the result of eachtoss.

According to my results I recorded seven trials with heads. Thiswould yield a relative frequency 7/10 or 0.7.

We call this type of computation an empirical probability.

When I toss the coin 100 times, my relative frequency was52/100 or 0.52. This number is closer to the theoreticalprobability, but it is not exact.

The relationship between relative frequencies and theoreticalprobabilities is called the Law of Large Numbers.

This law states that if you repeat the experiment a large numberof times, then the relative frequency of the outcome will tend tobe close to the theoretical probability of that outcome.


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Example 1

Find the probability and the odds of

drawing the given cards

1. A black card

2. A heart

3. A two of clubs

4. A queen

5. A face card

6. A card above 9 (count ace as high)


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Solution to example 1

Remember there are 52 cards in the deck. 26 are red, 26 are black.There are 4 suits diamonds (red), hearts (red), clubs(black),spades(black). There are 13 ranks in each suit 2, 3, 4, 5, 6, 7,8, 9, 10, J, Q, K, A. The face cards are the J, Q, K.

1. The probability of drawing a black card is 26/52=1/2. The odds

of drawing a black card is 26:26 or 1:1.2. The probability of drawing a heart is 13/52=1/13. The odds ofdrawing a heart is 13:39=1:3.

3. The probability of drawing the two of clubs is 1/52. The odds ofdrawing the two of clubs is 1:51.

4. The probability of drawing a queen is 4/52=1/13. The odds of

drawing a queen is 4:48=1:12.5. The probability of drawing a face card is 12/52=3/13. The odds

of drawing a face card is 12:40=3:10.

6. The odds of drawing a card above 9 is 20/52=5/13. The oddsof drawing a face card is 20:32=5:8.


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Example 2

Three coins are tossed answer the following:

1. What is the sample space?

ANSWER

S={HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}.

Define the following events E is the event of tossingexactly two heads. F is the event that there are at least2 are heads. G is the event that all three are heads.

2. What is P(E)? ANSWER n(E)/n(S)=3/8

3. What is O(E)? ANSWER n(E):n(E)=3:5

4. What is P(F)? ANSWER n(F)/n(S)=4/8=1/25. What is O(F)? ANSWER n(F):n(F)=4:4=1:1

6. What is P(G)? ANSWER n(G)/n(S)=1/8

7. What is O(G)? ANSWER n(G):n(G)=1:7


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Tossing 2 dice

Consider tossing 2 six-sided dice. What is the sample space?

We can enumerate the sample space by using ordered pairs like (x, y).X is the result of die 1 and y is the result of die 2.

Below is the sample space

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)

(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)

(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

We see that there are 36 outcomes in this sample space.

Suppose we add the faces together (which is what you do in mostgames that involve dice).

What is the probability of tossing a 7? ANSWER: The outcomes thatcorrespond to this event are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Thereare 6 outcomes so the probability is 6/36.


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More with two dice.

Define E as the event of tossing 2 dice such

that the sum of the faces is even.

What is P(E)? ANSWER n(E)/n(S)=18/36=1/2.

What is O(E)? ANSWER n(E):n(E)=18:18=1:1.

Define F as the event of tossing 2 dice such

that the sum of the faces is greater than 9.

What is P(F)? ANSWER n(F)/n(S)=6/36=1/6.

What is O(F)? ANSWER n(F):n(F)=6:30=1:5.

basic terms of probability dalesandro

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