An EVENT in statistics is an outcome or a set of outcomes of a random phenomenon/process/experiment. For example, an event might be "three correct answers in a row" if you were completely guessing on a multiple choice exam. An event might be "will vote for Gore". EVENTS have probabilities associated with them. You might like to think of a probability as a "chance".
PROBABILITY is the study of CHANCE: a certain random process is given (such as rolling a die or spinning a roulette wheel), and we want to know the chance of various outcomes (e.g. seeing a "6" on the die, seeing the ball land on a "red").
Note this definition implies the basic process if replicable; if the process cannot be repeated, using the word "chance" is not correct.
the number of outcomes of interest
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total possible number of outcomes
Independence can be thought of as when 2 events occur and one has no effect on the other occurring and vice versa. In other words, (from page 230) if A happens, it does not change the chance of B happening.A box with 5 tickets in it labeled 1,2,3,4,5
(a) if I pull out a ticket, what is the chance that it is a 4? (answer: 1/5)
(b) if I put the ticket back in and pull another out, what is the chance that it is a 3? (answer: 1/5)This is independence. Contrast it with this situation:
A box with 5 tickets in it labeled 1,2,3,4,5(a) if I pull out a ticket, what is the chance that it is a 4? (answer: 1/5)
(b) if I DO NOT put the ticket back in and then pull another out, what is the chance that it is a 3? (answer: 1/4 note that the chance of has changed) selecting a 3 is not independent. It is dependent where selecting a 4 first had some effect on selecting a 3 afterwards.You might think of this situation as selecting a ticket and then not replacing it PUTS A CONDITION on the next ticket selected.
This is a situation where you are trying to figure out the chances of two or more things happening. There is a sense of sequence here, something happens first, then something else occurs. It works for both situations of independence and of conditional probabilties.The chance that two things will happen equals the chance that the first will happen multiplied by the chance that the second will happen given that the first has happened.
In the situation where outcomes are independent, you multiply unconditional probabilities. Again, independence suggests unchange probabilities from outcome to outcome. If the outcomes are dependent (i.e. situations like sampling without replacement) then multiply conditional probabilities (changed probabilities in light of previous events).
This is a situation where you are trying to figure out the chance of two or more things happening BUT they cannot happen at the same time. If one happens, the other cannot happen, they are mutually exclusive events.MUTUALLY EXCLUSIVE -- when the occurrence of one prevents the occurrence of the other.
Example. A 90% of high school atheletes do not compete in college, about 10% do. Of those who compete in college, about about 2% make it to the pros. 98% do not. Of the 90% who do not compete in college about .1% make it to the pros. 99.9% do not. What percentage of ALL high school atheletes eventually make it to the pros?
First, you need to figure out who makes it to the pros of those who go to college and those who don't.
(.10 x .01) for those who went to college = .001 or .1% (.90 x .001) for those who did not go to college = .0009 or .09%
Why multiply? These are a sequence of events that are independent.
Then you need to add these two together. .1% + .09% = .19% Why add? Because these are mutually exclusive events -- if you did not go to college and then went to the pros -- you can't go to college and go to the pros. These are two different "tracks" leading to the same outcome but you want the total.
For some problems it might help you to list ways and it may help you to understand calculating chances.The atheletes again. There are 4 possible outcomes in life:
goes to college then goes pro
goes to college then does not go pro
does not go to college then goes pro
does not to college then does not go pro
The chances for each:
(.10)*(.01)=.001
(.10)*(.99)=.099
(.90)*(.001)=.0009
(.90)*(.999)=.8991
Quiz...what's the total outcome? 1.0 or 100%
If you toss a fair coin, you have two possible outcomes.
The chance of the coin landing heads up is 1/2.
The chance of the coin landing tails up is 1/2.
Let's move to a situation where you are tossing a coin 3 times:
There are 8 possible outcomes (2x2x2):
HHH HHT HTH HTT THH TTH THT TTT
When asked "what is the chance of getting 3 heads?" By listing the outcomes it's easy to see that it's 1/8. And if asked, "what is the chance of not getting 3 heads" it's easy to see that it's 7/8.
But listing outcomes may not be optimal it a situation like this one: you have 5 employees in your store. Employee A shows up 90% of the time, B shows up 75% of the time, C shows up 50% of the time, D shows up 90% of the time, and E shows up 60% of the time. What is the chance that at least one employee will be absent on a given day?
Try listing the ways....(don't)
Instead, you might think about:
Chance at least one absent = 100% - (chance of all 5 not absent)
example: I have a deck of cards, I deal a card to you, it can be a heart or it can be a spade, but it can't be both.What is the chance of getting dealt a heart? 13/52
What is the chance of getting dealt a spade? 13/52
example: What is the chance that the card I deal to you is a heart or a spade? 13/52 + 13/52 = 26/52 = 50%contrast it with: I deal a card to you. What is the chance that it is a heart? 13/52. I put the card back and deal another card to you, what is the chance that it is a spade? 13/52.
contrast it with: I deal two cards. I turn over the first card and it is a heart. What is the chance that the second one is a spade? 13/52 * 13/51. (Conditional probabilities)
NOTE: under INDEPENDENCE, A & B happen together, they just don't affect each other. Think of rolling dice.