Download Geometry: Statistics 12.6

Geometry: Statistics 12.6
A random variable is a variable whose value(s)
is determined by the outcomes of a
probability experiment
A probability distribution is a table which lists
the possible values of a random variable
along with the probability of each.
Note: The sum of the probabilities must be
____.
1
Example: Two 4-sided dice are rolled. Determine the
probability distribution for the sum of the two dice.
2
1
16
3
2
16
4
3
16
5
4
16
6
3
16
7
2
16
8
1
16
What is the probability the sum is greater than 4?
5,6,7,8
4 3 2 1 10
   
 .625
16 16 16 16 16
What is the probability the sum is a prime number?
2,3,5,7
1 2 4 2
9
   
 .5625
16 16 16 16 16
What is the probability the sum is greater than 4 or a prime
number?
2,3,5,7
5,6,7,8
1 2 4 3 2 1 13
     
 .8125
16 16 16 16 16 16 16
3 13
1 
 .8125
16 16
A special type of probability distribution is a binomial
distribution which results from a binomial probability
experiment. A binomial experiment must meet the
following criteria:
1. There are n independent trials
2. Each trial has only two possible outcomes : success or failure
3. The probability of success is the same for each trial
P(success)  p
P(failure)  1 - p
Example: Decide if the following situations satisfy the
conditions for a binomial distribution. If so, state n and p. If
not, state which condition is not satisfied.
A student randomly guesses the answer to each of ten multiple
choice questions that have 4 possible answers each.
It is binomial : n  10 and p  .25
A student pulls a card from a deck, notes if it is a heart, sets it a
side and draws another card. The student does this 8 times.
It is not binomial : probability on each trial changes
13 13or12 13or12or11
,
,
52 52
52
For a binomial experiment, the probability of
getting exactly k successes in n trials is given
by the formula:
P(k successes) = n Ck  p  1  p 
k
k
Example: If 25 coins are dropped on the floor, find
the probability that exactly 13 of them are heads.
Binomial
n  25
p  .5
k  13
C13  .5  1  .5
13
25
.155
12
Example: You are totally unprepared for a 10-question multiple
choice quiz, so you randomly guess on each question. If each
question has 5 possible answers, find P(get at least 8 questions
correct)
C8  .2  1  .5  .00007
8
Binomial
n  10
p  .2
k  8,9,10
10
2


C

.
2

1

.
5
 .000004
10 9
9
C10  .2  1  .5  .0000001
.0000741
10
10
1
0