Chapter 6, Problem 1UTA

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. $0\le r\le 31.$ Using appropriate computer software or calculators available to you. find the probability P(r) for each of the listed locations when r = 0, 1, 2…., 31.

To determine

The probabilities of $r=0,1,2,\cdots ,31$ for each of the locations.

Answer to Problem 1UTA

Solution: The probabilities for different cities are tabulated below:

Location	$P\left(r\right)$, $r=0,1,2,\dots ,30,31$
Juneau, Alaska	$0.0021,0.0145,0.0477,\dots 0,0.$
Seattle, Washington	$0.0002,\mathrm{0.002.0.0094},\dots ,0,0.$
Hilo, Hawaii	$0.00,0.00,0.0001,\dots 0,0.$
Honolulu, Hawaii	$0,0,0,\dots 0,0$.
Las Vegas, Nevada	$0,0,0,\dots ,0.0014,0.0001$
Phoenix, Arizona	$0,0,0,\dots ,0.0028,0.0003$

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, $n=31$.

The location, Juneau, Alaska, has 18% of clear days in December. So, the probability of success in a single trial (p) = 0.18 and $n=31$.

Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Juneau, Alaska, when $r=0,1,2,\cdots ,31$.

Step 1:

Press the $\boxed{\text{2nd}}$ button on the TI-83 Plus calculator and then press the $\boxed{\text{DISTR}}$ key.

Step 2:

Scroll down to binompdf (n, p, r) and press $\boxed{\text{ENTER}}$.

Step 3:

Enter the values of n, p and r as 31, 0.18 and 0, respectively, and press ENTER.

The value of $P(r=0)$ is approximately 0.0021.

Similarly, the rest of the probabilities for $r=1,2,3,\cdots ,31$ can be calculated by following the same steps as above, but changing the value of r.

Thus, the required probabilities are $0.0021,0.0145,0.0477,0.1012,0.1556,\dots ,\mathrm{0.000.}$

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 $n=31$.

Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Seattle, Washington, when $r=0,1,2,\cdots ,31$.

Step 1:

Press the $\boxed{\text{2nd}}$ button on the TI-83 Plus calculator and then press the $\boxed{\text{DISTR}}$ key.

Step 2:

Scroll down to binompdf (n, p, r) and press $\boxed{\text{ENTER}}$.

Step 3:

Enter the values of n, p and r as 31, 0.24 and 0, respectively, and press ENTER.

The value of $P(r=0)$ is approximately 0.0002.

Similarly, the rest of the probabilities for $r=1,2,3,\cdots ,31$ can be calculated by following the same steps as above, but changing the value of r.

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 $n=31$.

Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Hilo, Hawaii, when $r=0,1,2,\cdots ,31$.

Step 1:

Press the $\boxed{\text{2nd}}$ button on the TI-83 Plus calculator and then press the $\boxed{\text{DISTR}}$ key.

Step 2:

Scroll down to binompdf (n, p, r) and press $\boxed{\text{ENTER}}$.

Step 3:

Enter the values of n, p and r as 31, 0.36 and 0, respectively, and press ENTER.

The value of $P(r=0)$ is approximately 0.0000.

Similarly, the rest of the probabilities for $r=1,2,3,\cdots ,31$ can be calculated by following the same steps as above, but changing the value of r.

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 $n=31$.

Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Honolulu, Hawaii, when $r=0,1,2,\cdots ,31$.

Step 1:

Press the $\boxed{\text{2nd}}$ button on the TI-83 Plus calculator and then press the $\boxed{\text{DISTR}}$ key.

Step 2:

Scroll down to binompdf (n, p, r) and press $\boxed{\text{ENTER}}$.

Step 3:

Enter the values of n, p and r as 31, 0.60 and 0, respectively, and press ENTER.

The value of $P(r=0)$ is approximately 0.0000.

Similarly, the rest of the probabilities for $r=1,2,3,\cdots ,31$ can be calculated by following the same steps as above, but changing the value of r.

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 $n=31$.

Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Las Vegas, Nevada, when $r=0,1,2,\cdots ,31$.

Step 1:

Press the $\boxed{\text{2nd}}$ button on the TI-83 Plus calculator and then press the $\boxed{\text{DISTR}}$ key.

Step 2:

Scroll down to binompdf (n, p, r) and press $\boxed{\text{ENTER}}$.

Step 3:

Enter the values of n, p and r as 31, 0.75 and 0, respectively, and press ENTER.

The value of $P(r=0)$ is approximately 0.0000.

Similarly, the rest of the probabilities for $r=1,2,3,\cdots ,31$ can be calculated by following the same steps as above, but changing the value of r.

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 $n=31$.

Follow the steps given below in a TI-83 Plus calculator to obtain the probabilities for the location, Phoenix, Arizona, when $r=0,1,2,\cdots ,31$.

Step 1:

Press the $\boxed{\text{2nd}}$ button on the TI-83 Plus calculator and then press the $\boxed{\text{DISTR}}$ key.

Step 2:

Scroll down to binompdf (n, p, r) and press $\boxed{\text{ENTER}}$.

Step 3:

Enter the values of n, p and r as 31, 0.77 and 0, respectively, and press ENTER.

The value of $P(r=0)$ is approximately 0.0000.

Similarly, the rest of the probabilities for $r=1,2,3,\cdots ,31$ can be calculated by following the same steps as above, but changing the value of r.

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