Solutions to Worksheet for Assignment 7.4
1. a. The probability that the airline can accommodate everyone is the probability that 100 people or
fewer show up for the flight, or
. This is found by adding all the probabilities for the values
less than or equal to 100:
b. The probability that not everyone can be accommodated can be found two different ways. Since we
already calculated the probability of everyone being accommodated, we can just subtract that from 1.
Alternatively, we could add all the probabilities for values above 100, which would give the same result.
c. Being number 1 on the standby list means, if not all 100 people show up, you are the first person who
can be added to the flight. The probability of getting on would be the probability that less than 100
people show up.
Being number 3 on the list means that if no more than 97 people show up, you can get on the flight.
2. a. The easiest way to be sure you have all the possibilities for the four randomly selected
homeowners is with a tree diagram, with one set of branches for each homeowner. We can assume
that they are independent of each other, so the probabilities for having and not having insurance will be
the same for each.
S SSSS
S
F SSSF
S
F
S SSFS
S
S
S
F
F
S
SFSF
SFFS
S
F
S
SFFF
FSSS
F
S
F
S
FSSF
FSFS
S
F
F
F
S
F
F
F SSFF
SFSS
FSFF
FFSS
F
S
FFSF
FFFS
F
FFFF
For each set, P(S)=.2 and P(F)=.8. Multiply to get the probability of each branch set. The possible values
that X can take are the total number of S’s in each set, or 0, 1, 2, 3, or 4. Notice there is only one
outcome of 0 S’s and only one outcome of 4 S’s. There are four ways to get 1 S and four ways to get 3
S’s. Then there are six ways to get exactly 2 S’s. Add the probabilities for the like ways to get the
probability distribution.
X
0
1
2
3
4
P(X)
.4096
.4096
.1536
.0256
.0016
b. The most likely value of X is either 0 or 1, as they each occur with probability .4096 (the largest
probability). The EXPECTED value, which is the mean, is
.
c.
3. a.
This says that, in the long run, the salesperson averages 4.12 computer systems sold per month.
√
b.
√
This says that, on average, the number of computer systems sold per month by this salesperson varies
by about 1.4 systems.
c. The probability that the number of systems sold is within one standard deviation of the mean is
given by:
However, this is a discrete distribution, so we will use the values that X can take that still satisfy the
inequality:
Notice that since this is NOT a Normal distribution, we can NOT assume 68% of the probability is
within one standard deviation of the mean.
d. The probability that the number of systems sold is more than two standard deviations from the mean
is given by:
(
)
(
)