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Chapter
10 - Probability (Notes) Mr. Price (2005-2006)
1. What is probability? trying to find the chances of something happening
or .... finding out the likelihood of an event taking place
key words: probability, probable, likely, likelihood, chances,
possibilities, possible
2. If we know that an event is going to happen, we call that “certain” or
“guaranteed”.
The probability of a certain event is 1 (or 100%).
If we know that an event will never happen, we call that “impossible”.
The probability of an impossible event is 0 (0%).
You can’t have a negative probability or a probability over 100%.
3. Where is probability used? (these aren’t all the places)
a. Weather - trying to predict the chances of getting rain or having a
tornado
b. Stock Market - trying to predict if certain stocks (companies) will
rise in value
c. Gambling
1. Las Vegas - casinos
2. Lottery - keno, Powerball, etc.
d. Insurance - remember, insurance is all based on statistics!
1. car insurance - chances of having an accident
You must pay a “premium” to make sure you are covered.
Your premium depends on age, male/female, type of car, state you
are in,
grades, family driving history, smoker, etc.
2. house insurance - chances of getting hit by a tornado, having a
fire, etc.
Your premium depends on size of house, what it’s made of, how
much it is worth, how close it is to a fire department, etc.
3. life insurance - chances of you dying
4. health insurance - chances of you having to go to the hospital and
having
an operation, or getting sick and needing medical attention
5. There are all kinds of other insurance! If you want a good job,
become
an “actuary”, a person who works for the insurance company to do
all the math involved.
4. Why should we study probability? To understand those kinds of situations
listed above,
especially the gambling (negative) and insurance (positive).
5. How do you find the probability of something happening? How do you come up
with the
numbers? Note: Probability is something you haven’t studied much of and
it works
quite different than any of the other units we have covered. Be careful
and take good
notes.
a. More than likely, you will start with finding a fraction. You can
change the fraction
to a decimal and a percent if you want.
6. What is experimental probability? (Section 10-2)
Sometimes you can do an experiment to simulate what might take place.
Remember,
we can’t be for sure of the future, but we try to use math to help us make
a prediction.
For instance, in a baseball game, how likely is it that someone gets a
hit? Well, if that
particular person has never gotten a hit, ever, then it is unlikely. If
they have always
hit the ball, it is very likely (not quite certain) that they’ll get a hit.
Let’s say someone has gotten a hit 10 out of 15 times up to the plate. We
say they
have a 10/15 or 2/3 chance of getting a hit. That’s
Let’s say a car insurance company finds that 245 young men in Nebraska have
an
accident every year out of 10,000 young men who drive a car. Then
245/10,000 (or
49/2,000 in reduced terms) or 2.45% will probably have an accident.
7. Theoretical probability - trying to just think out probability but it’s not
related to anything
specific in real life.
Let’s take dice for instance (singular - die, plural - dice)
What are your chances of rolling a die and coming up with a “6”?
Your first question: how many sides does a die have? Answer: 6 (we
will
be using the regular dice).
Your second question: how many of those sides have a “6” on them?
Answer: 1
Therefore, your answer is 1/6 (a fraction). Now, you can change that
to a
decimal if you want (divide 1 by 6 to get 0.16666666) or a percent
(move
2 places over to get 16.666667%).
In math terms, we write P(6) = 1/6
Section 10-3
8. The best place to start is to create a “sample space”. A sample space is
just a fancy name
to say all the outcomes (possibilities).
For instance, the sample space for a die is only 6 (1, 2, 3, 4, 5, or a 6)
because a die
only has six faces (or sides). There are no more possibilities for one
die.
When we hit two dice, we have LOTS more possibilities. There are 36
possibilities
because each one on the first die can be paired up with each one on the
second die.
(6 * 6) The most common error in probability occurs here. Some kids think
there are
only12 because there are six on the first and six on the second, thus they
add. The key
to probability is actually multiplying!
Let me prove it.
(1,1) (2,1) (3,1) (4, 1) (5,1) (6,1)
(1,2) (2,2) (3,2) (4, 2) (5,2) (6,2)
(1,3) (2,3) (3,3) (4, 3) (5,3) (6,3)
(1,4) (2,4) (3,4) (4, 4) (5,4) (6,4)
(1,5) (2,5) (3,5) (4, 5) (5,5) (6,5)
(1,6) (2,6) (3,6) (4, 6) (5,6) (6,6)
The first question that is usually asked at this point is: “Isn’t (3,1)
the same thing
as (1,3)?” The answer: not in this case. In some instances, the order
does matter
and sometimes it doesn’t matter. In this case, (3,1) is a different
possibility than a
(1,3). If you were trying to do the comination on your locker, it
certainly does matter
what order you put the numbers in. If you were trying to dial a telephone
number,
it definitely matters what order you put the numbers!
*Note: The number of possibilities in your sample space is very, very
important.
It’s going to be the denominator of your fraction!
Let’s go back to a die.
P(prime number)? Well, there are only six sides on a regular die, so the
denominator is 6.
How many prime numbers are on a die? Well, 2, 3, and 5 are prime (1 is
neither prime
nor composite). Thus the P(prime number) is 3/6 or 1/2.
What has a better chance, getting a 1 or a 6? Sometimes kids think that
the six has a better
chance since it’s a bigger number. Wrong thinking. It turns out that they
have the same
probability because there’s only one 1 and one 6 on a same.
Back to dice ...
What is the probability of getting a sum of 2 when rollling two dice?
First question: how many possibilities are in the sample space? Answer:
36 (above)
Second question: how many of those possibilities will be a sum of 2?
Answer: Only 1.
How do I know? Well, the only way to get a sum of 2 is if you get a
(1,1). There is
no other way. Therefore, the probability is 1/36.
What is the probability of getting a product bigger than 30 when rollling
two dice?
First question: how many possibilities are in the sample space? Answer:
still 36!
Second question: how many of those will multiply to get an answer bigger
than 30?
Well, (6,6) will give 36. A (6,5) and a (5,6) will give exactly 30, but
it’s not BIGGER
than 30. P(product bigger than 30) = 1/36.
9. Coins
Another great way to talk about probability is to talk about coins.
One coin is easy: What is the P(heads) on one coin? Answer: 1/2
Why? There are two sides and only one of them is a heads.
Two coins starts getting a little tricky.
What is the probability of flipping two heads in a row? (H, H)
Answer: 1/4 because there are four answers in the sample space.
Here they are: (H, H) (H,T) (T, H) and (T, T). Remember (T, H) and
(H, T)
are not the same thing.
Only one of those possibilities is (H, H) so the answer is 1/4.
What is the chance of getting three heads in a row? Answer: 1/8
Here’s the listing: (H, H, H) (H, H, T) (H, T, H) (H, T,
T)
(T, T, T) (T, T, H) (T, H, T) (T, H, H)
Only one of those is all heads, so it’s 1/8.
Note: You could just multiply 2 * 2 * 2 to get the 8 instead of writing
them all out!
Section 10 - 4
10. Cards
Another great way to think about probability is to work with a regular deck
of cards.
There are 54 cards in a regular playing deck. Four suits of cards (hearts,
diamonds,
clubs, spades) plus 2 Jokers. I will tell you if it’s a regular deck or
without jokers,
since it is very important to know how many cards are in your sample space.
Take a regular deck of cards:
P(K) = ? Well, there are 54 cards. How many Kings are there?
Answer: There are 4 of them (one in each suit). Thus, you have 4/54
chance, or 2/27.
P(Q) = ? Answer: same as the chances of getting a King, 2/27.
P(Red card) = ? Well, how many red cards are there? Answer: 13
diamonds and
13 hearts. So there are 26 red cards you could possibly get, so 26/54,
which reduces
to 13/27.
P(numbered card) = ? How many numbered cards are there? (There are no
1’s and we
don’t count an Ace as 1 even though a lot of games count the Ace as 1
point). Let’s see..
2, 3, 4, 5, 6, 7, 8, 9, and 10 in four suits. 9 * 4 suits gives us 36
cards. Answer: 36/54
or 18/27 or finally 2/3. That’s a pretty good chance!
11. Getting two in a row
The probability gets a little tougher when we start taking out two cards in
a row
(or three, or four, etc.). Let’s say we want to know what the probability
would be
of getting two Kings in a row.
P(K, K) = ? Take the first card. There are 4 Kings to choose from, so
it’s 4/54 chance,
reducing to 2/27.
Take that card out of the deck (you just selected it and pulled it out of
the hat). What’s
the chances that the next card, by itself, is a King? Well, there are only
3 Kings left
(you took one out!), so it’s 3/53 (only 53 because you took one card out
of the pile).
P(K, K) = 2/27 times 3/53 or 2/477 (do your cross-canceling correctly)
P(Red, Black) = ? Take the Red first ..... 26/54 (there are 13 cards
in every suit,
plus the two Jokers). 26/54 reduces to 14/27.
Now find the probability of the the Black card. There are still 26 cards
that are Black
since we took out only a Red card. But ... there are only 53 cards left in
the deck.
So ... 26/53.
What’s the chances of drawing them both out, with the Red card first
followed by
the Black card?
P(Red, Black) = 14/27 * 26/53 or 364/1431 (you could make that into a
decimal or
a percent if you wish).
P(prime numbered card) = ?
Well, how many prime numbered cards are there? 2, 3, 5, and 7 in each
suit, so 4 * 4
or 16 cards. Put 16 over 54 to get 16/54 or 8/27.
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