Download binomial random variable

AP Statistics
• Section 8.1: The Binomial Distribution
• Objective: To be able to understand and
calculate binomial probabilities.
• Criteria for a Binomial Random Variable:
1. Each observation can be classified as a success or
failure.
2. n is the number of trials and n is fixed.
3. p is the probability of success and p is fixed.
4. The observations are independent.
If data are produced in a binomial setting and X = the
number of successes, then X is called a binomial random
variable.
The binomial probability distribution is as follows:
•
𝑿 = 𝒙𝒊
0
1
2
…
n
P(𝑿 = 𝒙𝒊 )
𝒑𝟎
𝒑𝟏
𝒑𝟐
…
𝒑𝒏
F(𝑿 = 𝒙𝒊 )
𝒑𝟎
𝒑𝟎 + 𝒑𝟏
𝒑𝟎 + 𝒑𝟏 + 𝒑𝟐
…
1
Points:
• A binomial random variable is always discrete.
• To graph use a probability histogram.
• Notation: X~B(n,p)
• When the population > 10n (sample size) and the sample
is selected without replacement we have an approximate
binomial distribution. It is approximate because the
condition of independence is violated. Since the effect is
minimal, we can still use the binomial formula for
calculations.
Ex. Let p = 0.3, Population (N) = 50,000, n = 5
• Binomial Probability Formula:
• Let X be a binomial random variable and k be the
number of successes. If X~B(n,p), then
𝑛 𝑘
• 𝑃 𝑋=𝑘 =
𝑝 ∙ (1 − 𝑝)(𝑛−𝑘) where
𝑘
𝑛
𝑛!
•
=𝑛 𝐶𝑘 =
which is called the binomial
𝑘! 𝑛−𝑘 !
𝑘
coefficient.
• (1 - p) = q
q represents the probability of failure.
𝑛 𝑘 (𝑛−𝑘)
• 𝑃 𝑋=𝑘 =
𝑝 ∙𝑞
(revised with q)
𝑘
Ex. A large shipment of toys contains 30% that are red,
40% that are blue and 30% that are yellow. Let X = the
number of red toys. You select 5 toys at random.
a. Does this example meet the criteria for a binomial
setting?
b. Find P(X = 0)
c. Find P(X = 1)
d. Find P(X = 2)
Calculator notation:
For P(X = k) --- use binompdf(n,p,X)
For P(X ≤ k) --- use binomcdf(n,p,X)
e. Create the probability distribution of X. Include the
cumulative distribution.
𝑿 = 𝒙𝒊
P(𝑿 = 𝒙𝒊 )
F(𝑿 = 𝒙𝒊 )
• F( X = x) is the cumulative density function.
𝐹 𝑋 = 𝑥𝑖 = 𝑃(𝑋 ≤ 𝑥𝑖 )
• The shape of the distribution of 𝐹 𝑋 = 𝑥𝑖 is always ________
• Which parameter, n or p, has the greatest influence in the
shape of the probability distribution?
• If p = 0.5 then the distribution is ____________________
• If p < 0.5 then the distribution is ____________________
• If p > 0.5 then the distribution is ____________________
• Ex. 7 couples buy new homes. It is known that in this
large community 24% of new homes are built with
electric heat. Let X = the number that choose electric
heat.
• Find the probability that 3 couples choose electric heat. (show the
formula that you would use)
• Find the probability that at most 1 couple chooses electric heat.
• Find the probability that at least 3 couples choose electric heat.
• Find the probability that more than 3 couples choose electric heat.
• Binomial Means and Standard Deviations
• If the random variable X has a binomial distribution with
n observations and the probability of success p, then the
mean and standard deviation of X are
• 𝜇 = 𝑛𝑝
and
𝜎 = 𝑛𝑝𝑞
•
•
•
•
•
Normal Approximation to the Binomial Distribution
Suppose 𝑋~𝐵(𝑛, 𝑝). When n is large,
𝑋~𝐴𝑝𝑝𝑟𝑜𝑥𝑁(𝑛𝑝, 𝑛𝑝𝑞).
Points:
Normal Approximation condition: Both 𝑛𝑝 ≥
10 𝑎𝑛𝑑 𝑛𝑞 ≥ 10
• If either condition is not met then the distribution is too
skewed to use a normal approximation.
• This method was used prior to the creation of calculators
and computers because as n gets larger and larger the
calculations become more time consuming.
• As n increases, the binomial distribution approaches a
normal distribution.
• Results for the normal approximation will be optimal
when n is large and p is close to 0.5
• We use an 𝐴𝑝𝑝𝑟𝑜𝑥𝑁 notation because it will never be
exactly normal because the binomial distribution is
discrete and the normal distribution is continuous.
Ex. Let X = number of successes; n = 100, p = 0.75.
a. Find the exact solution to P(X < 70)
b. Using the Normal Approximation, find the P(X < 70).
Ex. Suppose from previous exit polls it is known that the
probability that a voter will choose candidate A is 0.56. In
an upcoming election, you plan on asking 1000 randomly
selected voters who they will vote for. Using a Normal
Approximation, find the probability that between 501 and
550 voters inclusive will vote for candidate A.
(OPTIONAL)
Binomial Simulation:
Methods:
1. The book method: Use randbin(1, p, n). This will
choose the number 1 (for success) with probability p
and 0 (for failure) with probability q. This emphasizes
the binomial philosophy of success and failure. Then
you must sum the 1s to arrive at the total number of
successes.
2. Recommended method: Use randbin(n, p, number of
simulations). This will give you the number of
successes in n trials with a probability of success of p. It
will do this for as many simulations as desired.
3. Use the methods learned in chapter 5.
Ex. In the Hershey Kiss activity that we did in chapter 6, we
found that approximately 30% of the time a Hershey Kiss
lands point up. Suppose you flip a kiss 100 times. How
likely is it that the kiss to lands point up more than 35
times?
a. Use simulation to determine an approximate answer.
b. Use the binomial formula to find the exact answer.
c. Use the Normal Approximation to arrive at an
approximate answer.