Use the following situation for Problems 12 through 22. A couple plans to have no more than three

Use the following situation for
Problems 12 through 22.

A couple
plans to have no more than three children, and they will keep having children
until they have a boy. So, if their first child is a boy, they will stop and
have only one child. However, if their first child is a girl, they will try
again and have a second child. This time, we’ll assume that girls and boys are
equally likely.
Let X be the number of girls a couple
will have. Using the multiplication rule of independence,
P(0 girl)
= ½ (They had a boy on the first try.)
P(1 girl)
= ½ * ½ (They had a girl, then a boy.)
P(2
girls) = ½ * ½ * ½ (They had two in a row, then a boy.)

P(3 girls) = ______________ (They never had a boy! All three children
are girls)

1)
Find the missing probability. Check
to make sure your answers above sum to 1.
2)
Complete the table below using your work above.

Girls

0 girls

1 girl

2 girls

3 girls

Probability

3)
Find the expected number of
girls per couple following this plan.
4)
Find the total expected number
of girls among 1000 couples following this plan.
5)
Discussion: Do you think the expected number of girls
would be more than, less than, or the same as the expected number of boys? Explain your reasoning.

6)
Let C be the number of children a couple will have under the family
planning method above. Fill in the
missing probabilities.
P(C
= 1) = ½ (They had a boy on
the first try.)
P(C = 2) = ______ (They had a girl, then a boy.)
P(C = 3) =_______(They had two girls and a boy OR three girls. Hint: Add
two probabilities.)
7)
Complete the table:

Children

1 child

Probability

8)
Find the expected number of
children per couple.
9)
Find the total expected number
of children among 1000 couples following this plan.
10)
Would they have more girls, or
more boys or would it be the same?
(Hint: See above where you found
out the expected number of girls.) Is
this what you predicted in #16?
11)
Did we use theoretical or
empirical probabilities in this problem?

Use the following situation for
Problems 23 through 31.

Consider
the probability distribution of X,
where X is the number of times a
college graduate changed majors.

X

0

1

2

3

4

5

6

7

8

P(X)

0.135

0.271

0.271

0.180

0.090

0.036

0.012

0.003

0.002

12)
We collect a random sample of
1000 college graduates. Based on the probability distribution, which of the
following results would be surprising?
a) 130 graduates in the sample never changed their major.

b) 130 graduates in the sample changed their major 4
times.

c) 271 graduates in the sample changed their major
more than once.

13)
The mean of the distribution is
2.002 and the standard deviation is 1.419. What are “typical values” for the
number of times a college graduate changed majors?
14)
What percentage of the values is within one standard deviation of the
mean?
15)
Would it be unusual for a
college graduate to never have changed majors?
(Is this value more than 2 SD from the mean?)
16)
Which values for X are unusual?
17)
What percentage of the values
is unusual?
18)
What’s the probability that a
college graduate changed their major change at least once?
19)
What shortcut should you use to
answer the previous question? (You don’t
have to add 8 numbers!) What’s the name of the rule you used?

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