Download Probability Distributions - Somerville School District

Probability Distributions
Constructing a Probability Distribution
• Definition: Consists of the values a random
variable can assume and the corresponding
probabilities of the values. The probabilities are
determined either theoretically or by observations.
• Must meet 2 requirements:
• #1 The sum of the probabilities of all the events,
in the sample space, must be equal to 1.
• #2 The probability of each event, in the sample
space, must be between or equal to 0 and 1.
Prob. Distribution Examples
• The probability a tutor sees 1,2,3, or 4
students a day is .63, .20, .12, and .05
respectively.
• The probability of picking a heart, diamond,
spade, or club, when picking one card from a
deck.
• A box contains 3 $10 bills, 4 $50 bills, and 2
$100 bills. Construct a probability distribution
for the data.
Probability Distribution?
Yes or No
• Determine if the following actually are
probability distributions.
X
1
2
3
4
P(X)
⅕
⅕
⅖
⅖
X
5
10
15
20
P(X)
0.35
0.15
0.4
0.1
Means of Probability Distributions
• Can not simply use the old formula of finding
the sum and dividing by total number of
values. This formula would only approximate
the true mean.
• New Steps:
• #1 Multiply all possible outcomes by the
probability of each outcome.
• #2 Add all answers together.
• Formula: ∑ [ X ● P (X) ]
Mean Examples
• The number of pizzas sold in an hour at a local
pizzeria is shown in the table, with the
corresponding probabilities. Find the mean of
the data.
Pizzas Sold (X)
20
21
22
23
24
Probability, P(X)
0.1
0.25
0.2
0.3
0.15
Mean Examples
• The number of children living in a household,
in Somerville, is shown in the following table.
Find the mean of the data.
# of Children
0
1
2
3
4
Probability, P(X)
0.32
0.28
0.15
0.19
0.06
Variance and Standard Deviation of
Probability Distributions
• Step 1: Find the mean of the data, non rounded
answer, and square it. (To be used later)
• Step 2: Square each possible outcome and
multiply the answer to its corresponding
probability.
• Step 3: Find the sum of the answers of Step 2.
• Step 4: Step 3 answer – Step 1 answer. (Variance)
• Standard Deviation = √ Variance.
Variance and Standard Deviation
Examples
• Find the variance and standard deviation of
the data below.
# of Children
0
1
2
3
4
Probability, P(X)
0.32
0.28
0.15
0.19
0.06
Pizzas Sold (X)
20
21
22
23
24
Probability, P(X)
0.1
0.25
0.2
0.3
0.15
Expected Value
• Calculated in the same way as the mean for a
probability distribution.
• The symbol for expected value is E(X).
• In games of chance an expected value = 0 means
the game is fair. If expected value is positive then
the game is in favor of the player. If negative the
game is in favor of the house.
• Represents the average losses or gains that each
person can expect to have if they are part of the
game.
E(X) Examples
• Example 1: A local school is selling 500 tickets for a
raffle. Each ticket costs $5. The prizes are one $100
prize, one $50 prize and three $25 prizes. Find the
expectation if the person buys just one ticket.
• Example 2: A player rolls a die. If he/she gets a 1 or 6
they win $10. It costs $4 to roll the die. Find the
expectation.
• Example 3: Jenkinson’s boardwalk loses $60,000 a
season when it rains too much and makes $350,000
when it is sunny. The probability of too much rain over
the summer is 18%. Find the expectation for the profit.
Binomial Experiment
• A probability experiment that satisfies four
conditions:
• 1: Each trial can have only two outcomes or
outcomes that can be reduced to two outcomes.
These outcomes can be considered as either
success or failure.
• 2: There must be a fixed number of trials.
• 3: The outcomes of each trial must be
independent of each other.
• 4: The probability of a success must remain the
same for each trial.
Binomial Distribution
• The outcomes of the binomial experiment lead to
a special probability distribution.
• Some binomial notation:
• P(S) = probability of success
• P(F) = probability of failure
• p = the numerical probability of success
• q = the numerical probability of failure. ( 1 – p )
• n = number of trials
• X = the number of successes
Binomial Probability Formula
• In a binomial experiment, the probability of
exactly X successes in n trials is:
Binomial Comparison
• If a couple has 3 children what is the probability they
will have exactly 2 girls?
• Old method:
• 1: Use a tree diagram and find the sample space.
• 2: Determine how many outcomes have exactly 2
girls.
• Binomial method:
• 1: Use the formula shown before. Where n = 3, X = 2,
p = 1/2, q = ( 1 – p) =1/2
Binomial Examples
• Example 1: A student takes a 6 question, truefalse quiz and guesses on each question. Find the
probability of passing if the lowest passing grade
is at least 4 correct out of 6.
• Example 2: If someone tosses a coin 8 times find
the probability they will toss exactly 5 tails.
• If 30% of people in a community use the local
emergency room at a hospital in one year, find the
probability from a sample of 10 people that
exactly 3 used the emergency room.
Mean, Variance, and Standard Deviation
of the Binomial Distribution
•
•
•
•
Mean = n ● p
Variance = n ● p ● q
Standard Deviation = √ Variance
Example 1: A die is rolled 50 times. Find the
mean, variance, and standard deviation for the
number of 5’s that will be rolled.
• Example 2: A study found that 5% of Americans
work within 10 miles of their home. If 650
people are selected at random find the mean,
variance, and standard deviation of the number of
people who work within 10 miles of their home.