Download 6.3Part I The Binomial Distributions

6.3 Part I The Binomial Distributions
Ex 1: Suppose a family (with serious birth control issues) has 7 children. What is the
probability that exactly 4 of them are girls?
a) Can you figure out a way to use the table of random digits to carry out a
simulation that will give an approximation of this probability? Describe the
method but do not carry out the simulation.
b) Can you think of any other ways to carry out this simulation?
The Binomial Setting
1. Each observation falls into one of just two categories, which we call “success” or
“failure.” We let X = the number of successes (Discrete)
2. There is a fixed number of observations, which we call n.
3. The n observations are independent.
4. The probability of “success,” called p, is the same for each observation.
c) Does Example 1 satisfy the assumptions of a binomial setting? If it does, define X,
n, and p.
d) Use your calculator to find P(X=4). To find P(X=k), the command is
binompdf(n,p,X) under DISTR.
e) Find the probability that the family has at least 5 girls.
The Cumulative Binomial Setting
Ex 2: The probability that Daniel Sedin scores on a random shot on net is 0.36. In one
game Daniel shoots on net 15 times.
a) Is this a binomial setting? Why or why not?
b) Assuming that the situation is binomial, define X, n and p.
c) If we were asked to find the probability that at most 2 goals were scored by
Daniel, using binompdf would be quite time consuming. Why?
d) In situations involving P( X ≤ k) , it is useful to use the command binomcdf(n,p,k),
which accumulates probabilities up to and including k. Use this to answer (c).
e) Find P(X=4)
f) Find P(X>4)
g) Find P(X ≥ 2)
Mean and Standard Deviation of a Binomial Random Variable
As with all distributions, continuous or discrete, the binomial random variable has a
certain shape, given by
µ = np and σ = np(1 − p)
Find the mean and standard deviation for Example 2.