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Conditional and Unconditional Chance.
Conditional and Unconditional Chance
Conditional Chance puts conditions on a problem.
Unconditional Chance puts not conditional on the first
- draw two cards from a deck
- chances of drawing a queen from a card of deck is 1/52. (Unconditional)
- If first card is not a queen the chance turns to 1/51 (condictional)
Independent if the chances for the second given the first are the same, no matter how the first one turns out. Otherwise the two things are dependent
flipping a coin, rolling the dice are independent they don't depend on the first outcomes
drawing cards without replacement, the draw are dependent or mutually exclusive
- When drawing with replacement the draws are independent
- It was developed by Abraham de Moivre a French Protestant
- It was developed to solve gambling problems
- work best for process that can be repeated over and over again independently and under the same conditions
The Chance of Something
The Chance of something gives the percentage of time something is supposed to happen when the process is done over and over again, independent and under the same conditions
- Chances are between 0% and 100%
The multiplication rule
The chance that two things will both happen equals the chances that the FIRST will happen, multiplied by the chance that the second will happen given the first has happened.
The multiplication rule finds the chance that two things both happens
- what are the chances that some one will draw a 1 and then a 2 in a box that has three cards 1,2,3.
Hint: Look for or
- first chance is 1/3
- second chance is 1/2
1/3 x 1/2 =1/6
- Deck. If a deck of cards is there it is always 4 suits of the same card (diamond,spade,heart,club).
- If we wanted to know the draw chances of getting queen then we would say 4/52... The chances of the second draw being a queen is 4/51 given the first draw isn't a queen. If first draw is a queen then it would be 3/51.
(note: alway remember to find-out how many of a color, suits, items, etc is in the draw)
with dice and flipping the coin chances refresh each flip/roll of the dice regardless of previous roll or flip (it refreshes with each roll/flip)
- chances of getting a 6 the first a 3 the second is
1/6 x 1/6 =1/12
The addition rule
To find the chance that at least one of two thing will happen. Check to see if they are mutually exclusive. If the are add the chances.
The addition rule find the chance that at least one of two things happens
Look for Both or And
Two things are mutually exclusive
when the occurrence of one prevents the occurrence the other (one excludes the other)
- Mutually Exclusive
- If the card is an Spade it can't be a Diamond. The Spade prevents the diamond (ie the events can't both happen)
- Not Mutually Exclusive
- What are the chances of rolling 1 with two dice.
Both are independent one dice does not prevent the other.
P(B|A) = the conditional probability that event B occurs given that A occurs (probability of B given A") A standard deck of cards has
Counting Equally Likely Outcomes.
For equally likely outcomes, P(A) = (# of ways that A can occur)/(total number of possible outcomes)
Examples: a) the chance of drawing an ace from a deck of cards = 4/52
b) the chance that the sum of two die rolls is 4 = 3/36 (see p238-9 of text) 3 because there are 3 ways to get the sum of (1-3,2-2,3-1)
2. Chance of the Complement (Opposite).
P(A) = 1 - P(Ac)
Examples: a) If I draw a card from a deck of cards, the chance that it is not an ace is 1 - 4/52 = 4852
.b) I roll a fair six-sided die twice. The chance I get at least one six is 1 - the chance of no sixes, or 1 - (5/6)2.
(See 3a) below.) Hint: Look for problems which say at least one not not all" these have opposites which are easy to compute.
Multiplication Rule. P(A and B) = P(A) x P(B|A)
The book also discusses the special case when then events are independent. When A and B are independent, event B has the same chance regardless of whether event A happens, so the multiplication rule has the special form P(A and B) = P(A) x P(B)
Examples: a) I roll a fair six-sided die twice. The chance I roll no sixes is 56 x 56 = 2/3
b) I draw two cards from a standard deck. The chance that both cards are spades is 13/52 x 12/51.
Hint: Look for and" or both"
P(A or B) = P(A) + P(B) - P(A and B)
The book form of the addition rule only tells you what to do when the two events are mutually exclusive (ie the events can't both happen). When the two events are mutually exclusive, P(A and B) = 0
a) I draw a card from a standard deck. The chance that I get a king or queen is 4/52 + 4/52 = 8/52
Note that the card can't be both a king and a queen so P(A and B) = 0
.b) I draw a card from a standard deck. The chance it is a king or a spade is 4/52 + 13/52 - 1/52 = 16/52 .
Hint: Look for (or")
When to Add or Multiple
First decide whether the events are independent or mutually exclusive
- P(A or B)
- They are independent of one another. Multiply.
B) They are dependent Mutually Exclusive.
=When you add events are more likely to happen. (There's a higher probability)
- Example: what is the chance of you getting a (5) or a (6) on two roll of the dice?
- The probability is
1/6 (+or) 1/6 =2/12 - .1666 (about a 17 percent chance it happens)
= when you multiply two events are less likely to happen (the probability goes down)
Example what is the probability of you getting a A Queen top and a King on the bottom of a stack of cards with two draw
4/52 (x and) 4/51 = 16/2652 = .006 (no probability)