Download Odds And Expected Value

Probability And
Expected Value
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——
Probability –————————
Probability is the measure of how likely to
occur an event is. To calculate the
probability, use the following formula:
P = the number of ways an event can occur
the total number of possible outcomes
For Example –———————
—
There are 26 letters in the alphabet. Five of
the letters are vowels. So the probability of
choosing a vowel from the alphabet is:
P(vowels) = the number of vowels = 5
total letters in alphabet 26
Practice –————————
A number from 1 to 11 is chosen at random.
What is the probability of choosing an odd
number? Remember our formula:
the number of ways an event can occur
P = the total number of possible outcomes
Hint: There are 6 odd numbers and there are
11 numbers we could choose from.
P = 6 ÷ 11
Independent events ——————
If there is a jar with 10 maroon marbles and 10
white marbles in it, what is the probability you’ll
choose a maroon marble?
On your first try you drew a maroon marble. If you then
replaced the maroon marble and drew again, what’s the
probability that you will draw another maroon marble?
These are called independent events since
the two events (your first draw and your
second) do not affect one another.
Dependent events ——————
If, on your first try, you drew a maroon marble, kept it,
and then drew a second marble, what’s the probability
that your second marble is also maroon?
**Hint: the total number of marbles has changed!**
What about the probability for drawing a white
marble on your second try?
These are called dependent events.
The probability of your second
draw is dependent on what you
drew the first time.
Lotteries –————————
In lotteries you must pick the correct
numbers to win, but there are different
versions of the game.
Version 1: Order does NOT matter, numbers CANNOT repeat
Suppose that to win the lottery we must choose any three numbers 0-9 in
any order. The numbers cannot repeat. Let’s say we chose the numbers
6,9, and 4. As long as our three numbers are drawn in any order, we win!
To find the probability that we will win,
we have to multiply the probabilities that
each of our numbers will be chosen.
Lotteries –————————
On the first draw the probability that a 6,9, or 4 will be drawn is
numbers, 3 numbers can be chosen.
3
10
. Out of 10 possible
Let’s pretend a 4 was drawn the first time.
2
On the second draw the probability that a 6 or a 9 will be drawn is 9 . Because we can’t
repeat, the 4 that we drew on the first draw has to be excluded from this second draw
making our total possible numbers 9. The numerator is now 2 because there are only
two number choices that can occupy this space to make us winners.
A 6 was drawn on the second draw.
1
The probability that our last number, 9, will be chosen on the third draw is 8 . We have
already drawn two numbers so our total choices are now 8. There’s only one number left
that can be chosen that will make us a winner.
Lotteries –————————
We now multiply all the individual probabilities together to
determine the probability that all three of our numbers will be
chosen and we will win.
3 2 1
6
1
  

10 9 8 720 120
We can expect to win only 1 out of every 120 lotteries!!
Lotteries –————————
Version 2: Order DOES matter, numbers CAN repeat
Again, we can choose any number 0-9. The order that we put these
numbers in must match the order in which the lottery numbers are drawn.
The numbers can repeat. Let’s say we chose the numbers 2,8, and 2
again in that order.
On the first draw, the probability that a 2 will be drawn out of a total of 10 available numbers
is 1 .
10
Because our numbers can repeat, we have to replace the 2 so that our total possible numbers to
draw from is again 10.
The probability that we will now draw an 8 is
1
10 .
Again, we replace the number we drew. The probability that our final number will be a 2 is 1 out of 10 .
Lotteries –————————
We now multiply the probabilities together to determine
our overall probability of winning:
1
1
1
1



10 10 10 1000
Which lottery version would you rather play? The one
where order doesn’t matter and numbers cannot be repeated
or the one where order does matter and repeating is okay?
Why?
Practice –————————
How would you
calculate the
probability that you
would win if the
version of the lottery
game had rules that
order mattered and
you were NOT
allowed to repeat your
number?
1
10
1
9
1
8
The answer is:
1/720
Expected Value –—————
Expected Value is defined as the sum of
all possible values for a random variable,
each value multiplied by its probability of
occurrence.
For example, in the game Hog, a player rolls a dice. If the dice lands on a 1, the player
gets zero points. If it lands on any other number, the player gets one point for every dot
on the dice. So if I rolled a 4, I get four points.
To calculate the expected value of rolling a dice in Hog, you multiply the amount you
win by the probability of rolling a specific number. Each number on the dice has a
1/6 probability of being rolled. Calculate this for each potential outcome and add
them all together.
1
1
1
1
1
1 20 1
(0  )  (2  )  (3  )  (4  )  (5  )  (6  )   3
6
6
6
6
6
6
6
3
This means that if you roll the dice 10 times, your expected score will be about 33!
Expected Value –—————
Pretend I flip a coin. If it lands on heads you win $5. If it
lands on tails, you win nothing. What is the expected
1
1
value?
($5  )  ($0  )  $2.50
2
2
I flip the same coin. This time if it lands on heads you
win $5, but if it lands on tails, you win $3. What is the
expected value?
1
1
($5  )  ($3  )  $4.00
2
2
This time, you win $5 if the coin lands on heads, but if it lands
on tails, you owe me $6. Calculate the expected value.
1
1
($5  )  ($6  )  $0.50
2
2
If you play ten times, you
Can expect to lose $5.
Wrap it up –————————
 Remember
some odds depend on
what already happened, some odds
don’t
 Remember the odds of winning are
important
 But, the expected value is even more
important