Concept explainers
Although tables of binomial probabilities can be found in most libraries, such tables are often inadequate. Either the value of p(the probability of success on a trial) you are looking for is not in the table, or the value of n (the number of trials) you are looking for is too large for the table. In Chapter 7, we will study the normal approximation to the binomial. This approximation is a great help in many practical applications. Even so. we sometimes use the formula for the binomial probability distribution on a computer or graphing calculator to compute the probability we want.
Applications
The following percentages were obtained over many years of observation by the U.S. Weather Bureau. All data listed are for the month of December.
Location | Long-Term Mean % of Clear Days in Dec |
Juneau, Alaska | 18% |
Seattle. Washington | 24% |
Hilo. Hawaii | 36% |
Honolulu, Hawaii | 60% |
Las Vegas, Nevada | 75% |
Phoenix, Arizona | 77% |
Adapted from Local Climatological Data, U.S. Weather Bureau publication, 'Normals, Means, and Extremes' Table.
In the locations listed, the month of December is a relatively stable month with respect to weather. Since weather patterns from one day to the next are more or less the same, it is reasonable to use a binomial probability model.
Let r be the number of clear days in December. Since December has 31 days.
The probabilities of
Answer to Problem 1UTA
Solution: The probabilities for different cities are tabulated below:
Location | |
Juneau, Alaska | |
Seattle, Washington | |
Hilo, Hawaii | |
Honolulu, Hawaii | |
Las Vegas, Nevada | |
Phoenix, Arizona |
Explanation of Solution
Given: A table consisting of the long-term mean percentage of clear days in December for the different locations has been provided.
Calculation: Consider that r represents a random variable that is defined as the number of clear days. Since December has 31 days,
The location, Juneau, Alaska, has 18% of clear days in December. So, the probability of success in a single trial (p) = 0.18 and
Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Juneau, Alaska, when
Step 1:
Press the
Step 2:
Scroll down to binompdf (n, p, r) and press
Step 3:
Enter the values of n, p and r as 31, 0.18 and 0, respectively, and press ENTER.
The value of
Similarly, the rest of the probabilities for
Thus, the required probabilities are
r | P(r) |
0 | 0.0021 |
1 | 0.0145 |
2 | 0.0477 |
3 | 0.1012 |
4 | 0.1556 |
5 | 0.1844 |
6 | 0.1754 |
7 | 0.1375 |
8 | 0.0906 |
9 | 0.0508 |
10 | 0.0245 |
11 | 0.0103 |
12 | 0.0038 |
13 | 0.0012 |
14 | 0.0003 |
15 | 0.0001 |
16 | 0.00 |
17 | 0.00 |
18 | 0.00 |
19 | 0.00 |
20 | 0.00 |
21 | 0.00 |
22 | 0.00 |
23 | 0.00 |
24 | 0.00 |
25 | 0.00 |
26 | 0.00 |
27 | 0.00 |
28 | 0.00 |
29 | 0.00 |
30 | 0.00 |
31 | 0.00 |
The location, Seattle, Washington, has 24% of clear days in December. So, the probability of success in a single trial (p) = 0.24 and
Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Seattle, Washington, when
Step 1:
Press the
Step 2:
Scroll down to binompdf (n, p, r) and press
Step 3:
Enter the values of n, p and r as 31, 0.24 and 0, respectively, and press ENTER.
The value of
Similarly, the rest of the probabilities for
The required probabilities are:
r | P(r) |
0 | 0.002 |
1 | 0.002 |
2 | 0.0094 |
3 | 0.0286 |
4 | 0.0632 |
5 | 0.1078 |
6 | 0.1475 |
7 | 0.1663 |
8 | 0.1575 |
9 | 0.1271 |
10 | 0.0883 |
11 | 0.0533 |
12 | 0.028 |
13 | 0.0129 |
14 | 0.0053 |
15 | 0.0019 |
16 | 0.0006 |
17 | 0.0002 |
18 | 0.00 |
19 | 0.00 |
20 | 0.00 |
21 | 0.00 |
22 | 0.00 |
23 | 0.00 |
24 | 0.00 |
25 | 0.00 |
26 | 0.00 |
27 | 0.00 |
28 | 0.00 |
29 | 0.00 |
30 | 0.00 |
31 | 0.00 |
The location, Hilo, Hawaii, has 36% of clear days in December. So, the probability of success in a single trial (p) = 0.36 and
Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Hilo, Hawaii, when
Step 1:
Press the
Step 2:
Scroll down to binompdf (n, p, r) and press
Step 3:
Enter the values of n, p and r as 31, 0.36 and 0, respectively, and press ENTER.
The value of
Similarly, the rest of the probabilities for
The required probabilities are:
r | P(r) |
0 | 0.00 |
1 | 0.00 |
2 | 0.0001 |
3 | 0.0008 |
4 | 0.0031 |
5 | 0.0094 |
6 | 0.0229 |
7 | 0.046 |
8 | 0.0775 |
9 | 0.1115 |
10 | 0.1379 |
11 | 0.1481 |
12 | 0.1389 |
13 | 0.1142 |
14 | 0.0826 |
15 | 0.0526 |
16 | 0.0296 |
17 | 0.0147 |
18 | 0.0064 |
19 | 0.0025 |
20 | 0.008 |
21 | 0.0002 |
22 | 0.0001 |
23 | 0.00 |
24 | 0.00 |
25 | 0.00 |
26 | 0.00 |
27 | 0.00 |
28 | 0.00 |
29 | 0.00 |
30 | 0.00 |
31 | 0.00 |
The location, Honolulu, Hawaii, has 60% of clear days in December. So, the probability of success in a single trial (p) = 0.60 and
Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Honolulu, Hawaii, when
Step 1:
Press the
Step 2:
Scroll down to binompdf (n, p, r) and press
Step 3:
Enter the values of n, p and r as 31, 0.60 and 0, respectively, and press ENTER.
The value of
Similarly, the rest of the probabilities for
The required probabilities are:
r | P(r) |
0 | 0.00 |
1 | 0.00 |
2 | 0.00 |
3 | 0.00 |
4 | 0.00 |
5 | 0.00 |
6 | 0.00 |
7 | 0.00 |
8 | 0.001 |
9 | 0.0004 |
10 | 0.0012 |
11 | 0.0034 |
12 | 0.0084 |
13 | 0.0185 |
14 | 0.0357 |
15 | 0.0607 |
16 | 0.091 |
17 | 0.1205 |
18 | 0.1406 |
19 | 0.1443 |
20 | 0.1298 |
21 | 0.102 |
22 | 0.0696 |
23 | 0.0408 |
24 | 0.0204 |
25 | 0.0086 |
26 | 0.003 |
27 | 0.0008 |
28 | 0.0002 |
29 | 0.00 |
30 | 0.00 |
31 | 0.00 |
The location, Las Vegas, Nevada, has 75% of clear days in December. So, the probability of success in a single trial (p) = 0.75 and
Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Las Vegas, Nevada, when
Step 1:
Press the
Step 2:
Scroll down to binompdf (n, p, r) and press
Step 3:
Enter the values of n, p and r as 31, 0.75 and 0, respectively, and press ENTER.
The value of
Similarly, the rest of the probabilities for
The required probabilities are:
r | P(r) |
0 | 0.00 |
1 | 0.00 |
2 | 0.00 |
3 | 0.00 |
4 | 0.00 |
5 | 0.00 |
6 | 0.00 |
7 | 0.00 |
8 | 0.00 |
9 | 0.00 |
10 | 0.00 |
11 | 0.00 |
12 | 0.00 |
13 | 0.0001 |
14 | 0.0003 |
15 | 0.0009 |
16 | 0.0028 |
17 | 0.0074 |
18 | 0.0173 |
19 | 0.0356 |
20 | 0.064 |
21 | 0.1006 |
22 | 0.1372 |
23 | 0.161 |
24 | 0.161 |
25 | 0.1353 |
26 | 0.0937 |
27 | 0.052 |
28 | 0.0223 |
29 | 0.0069 |
30 | 0.0014 |
31 | 0.0001 |
The location, Phoenix, Arizona, has 77% of clear days in December. So, the probability of success in a single trial (p) = 0.77 and
Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Phoenix, Arizona, when
Step 1:
Press the
Step 2:
Scroll down to binompdf (n, p, r) and press
Step 3:
Enter the values of n, p and r as 31, 0.77 and 0, respectively, and press ENTER.
The value of
Similarly, the rest of the probabilities for
The required probabilities are:
r | P(r) |
0 | 0.00 |
1 | 0.00 |
2 | 0.00 |
3 | 0.00 |
4 | 0.00 |
5 | 0.00 |
6 | 0.00 |
7 | 0.00 |
8 | 0.00 |
9 | 0.00 |
10 | 0.00 |
11 | 0.00 |
12 | 0.00 |
13 | 0.00 |
14 | 0.0001 |
15 | 0.0004 |
16 | 0.0012 |
17 | 0.0036 |
18 | 0.0094 |
19 | 0.0216 |
20 | 0.0433 |
21 | 0.0759 |
22 | 0.1156 |
23 | 0.1514 |
24 | 0.1689 |
25 | 0.1584 |
26 | 0.1224 |
27 | 0.0759 |
28 | 0.0363 |
29 | 0.0126 |
30 | 0.0028 |
31 | 0.0003 |
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Chapter 6 Solutions
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- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,